\documentclass[reqno]{amsart} \usepackage{graphicx} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 34, pp. 1--20.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/34\hfil Biharmonic elliptic problems] {Some remarks on biharmonic elliptic problems\\ with positive, increasing and convex nonlinearities} \author[E. Berchio, F. Gazzola\hfil EJDE-2005/34\hfilneg] {Elvise Berchio, Filippo Gazzola} % in alphabetical order \address{Elvise Berchio \hfill\break Dipartimento di Matematica, Universit\'a di Torino, Via Carlo Alberto 10 - 10123 Torino, Italy} \email{berchio@dm.unito.it} \address{Filippo Gazzola \hfill\break Dipartimento di Matematica del Politecnico, Piazza L. da Vinci 32 - 20133 Milano, Italy} \email{gazzola@mate.polimi.it} \date{} \thanks{Submitted October 10, 2004. Published March 23, 2005.} \subjclass[2000]{35J40, 35J60, 35G30} \keywords{Semilinear biharmonic equations; minimal solutions;\hfill\break\indent extremal solutions} \begin{abstract} We study the existence of positive solutions for a fourth order semilinear elliptic equation under Navier boundary conditions with positive, increasing and convex source term. Both bounded and unbounded solutions are considered. When compared with second order equations, several differences and difficulties arise. In order to overcome these difficulties new ideas are needed. But still, in some cases we are able to extend only partially the well-known results for second order equations. The theoretical and numerical study of radial solutions in the ball also reveal some new phenomena, not available for second order equations. These phenomena suggest a number of intriguing unsolved problems, which we quote in the final section. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In the previous two decades, positive solutions to the second order semilinear elliptic problem $$\label{secondorder} \begin{gathered} -\Delta u=\mu g(u)\quad \mbox{in }\Omega \\ u=0\quad \mbox{on }\partial\Omega \end{gathered}$$ have attracted a lot of interest, see e.g.\ \cite {brezis,brezis2,brezis3,crandall,gazzola3,gelfand,Joseph,li,Martel,mignot} and references therein. Here, $\Omega$ is a smooth bounded domain of $\mathbb{R}^{n}$ $(n\geq 2)$, $\mu\geq0$ and $g$ is a positive, increasing and convex smooth function. By now, (\ref{secondorder}) is quite well understood. As a subsequent step, P.L.\ Lions \cite[Section 4.2 (c)]{li} suggests to study positive solutions to \emph{systems} of semilinear elliptic equations, namely $$\label{system} \begin{gathered} -\Delta u_i=\mu g_i(u_1,\dots ,u_m)\quad \mbox{in }\Omega \\ u_1=\dots =u_m=0\quad \mbox{on }\partial\Omega \end{gathered}$$ $(i=1,\dots ,m)$, where $m\ge2$ and the functions $g_i$ are as just mentioned. In this paper we consider the case of two equations ($m=2$) with $g_1(u_1,u_2)=u_2$ and $g_2(u_1,u_2)=g(u_1)$. Then, taking $\lambda=\mu^2$, system (\ref{system}) reduces to the following semilinear biharmonic elliptic problem under Navier boundary conditions: $$\begin{gathered} \Delta ^{2}u=\lambda g(u)\quad \mbox{in }\Omega \\ u=\Delta u=0\quad \mbox{on }\partial\Omega\,. \end{gathered} \label{general problem}$$ We will focus essentially our attention on the cases where $g$ is logarithmically convex, namely $$\begin{gathered} g\in C^1(\mathbb{R}_+)\,,\quad g(0)>0\,,\\ s\mapsto\log g(s)\quad \text{is nonconstant increasing and convex}, \end{gathered} \label{log-convex}$$ or $g$ has a power-type behavior such as $$g(s)=(1+s)^{p},\quad p>1\,. \label{power.type}$$ Very little is known about (\ref{general problem}) when $g$ satisfies (\ref{log-convex}) or (\ref{power.type}). As far as we are aware, only a couple of papers \cite{arioli,reichel} considering \emph{Dirichlet} boundary conditions study this problem. But it is well-known that boundary conditions significantly change the nature of the problem and of the tools available in the proofs. For instance, under Navier boundary conditions one has maximum and comparison principles in any domain $\Omega$. On the other hand, when dealing with Dirichlet boundary conditions one seeks solutions in $H^2_0(\Omega)$ and this allows one to extend solutions by $0$ \textit{outside} $\Omega$; see, in particular, Problem \ref{uniqureichel} in Section \ref{problems}. The first purpose of the present paper is to extend to (\ref{general problem}) some well-known results relative to (\ref{secondorder}). In Theorem \ref{log-convex-theo} we assume that the source $g$ satisfies (\ref{log-convex}) and we prove a full extension of the results available for (\ref{secondorder}). Although the results remain similar, the proof is completely different due to some technical difficulties, see Problem \ref{limitatezza} in Section \ref{problems}. We overcome this problem by generalizing a procedure developed in \cite{arioli}. Then, we turn to the power-like case (\ref {power.type}). When $p$ is subcritical, namely $p\le(n+4)/(n-4)$, by applying critical point techniques as in \cite{ar,brezis2,crandall,ge} in Theorem \ref{sottocritic.theo} we completely extend the results relative to (\ref{secondorder}). But for supercritical $p$, namely $p>(n+4)/(n-4)$, we only have partial results, see Theorem \ref{supercritic.theo}. The second (and perhaps most important) purpose of the present paper is to emphasize some striking differences between (\ref{secondorder}) and (\ref {general problem}). These differences are not just the already mentioned technical difficulties in the proofs but also some unexpected and new behaviors of the solutions which are particularly evident in the radial setting, i.e.\ the case where $\Omega=B$, the unit ball. Let us mention a couple of these differences. When $g(s)=e^s$ or $g(s)=(1+s)^p$ one can easily find explicit singular radial solutions of (\ref{secondorder}), see \cite{brezis3,mignot2}. For the same nonlinearities $g$, one can also find explicit singular solutions of the equation in (\ref{general problem}) which satisfy the first boundary condition but \emph{not} the second. Hence, apparently, these are ghost'' singular solutions which have nothing to do with problem (\ref{general problem}). But in \cite{arioli} it was shown that the true'' singular solutions have the same asymptotic blow up behavior as the ghost solutions. We have no explanation of this fact. If $g$ is critical, namely $g(s)=(1+s)^{(n+2)/(n-2)}$, problem (\ref {secondorder}) may be solved explicitly when $\Omega=B$, see \cite {gazzola3,Joseph}. Up to rescaling and translations, the solutions are the restrictions to $B$ of the positive entire solutions of the equation $-\Delta u=u^{(n+2)/(n-2)}$ over $\mathbb{R}^n$. For critical growth problems of fourth order, namely $g(s)=(1+s)^{(n+4)/(n-4)}$, the same result is not true. The reason is that Pohozaev identity \emph{does not} ensure nonexistence of radial sign changing solutions of $\Delta^2u=|u|^{8/(n-4)}u$ over $\mathbb{R}^n$, see Problem \ref{nodalradial}. With the aid of Mathematica we numerically show that the previous equation has both radial positive solutions which (for finite $|x|$) blow up towards $+\infty$ and solutions which change sign and (for finite $|x|$) blow up towards $-\infty$. Then, by a shooting method having the initial second derivative as parameter, in Theorem \ref{nuovo} we partially prove these numerical evidences. These are just some differences between (\ref{secondorder}) and (\ref {general problem}), for further differences see Section \ref{someremarks}. These surprising results shed some light on semilinear fourth problems but still much work has to be done to reach a complete understanding of (\ref {general problem}) and (\ref{system}). This leads us to suggest some (difficult) unsolved problems in Section \ref{problems}. The paper is organized as follows. In next section we establish our main results for general domains $\Omega$. In Section 3 we prove some analogies between (\ref{secondorder}) and (\ref{general problem}) for a wide classs of nonlinearities $g$. In Section 4 we study the particular case where $\Omega$ is the unit ball and we emphasize some differences between (\ref{secondorder}% ) and (\ref{general problem}). Sections 5-8 are devoted to the proofs of the results. Finally, in Section \ref{problems} we quote some open problems. \section{Main results} Throughout the paper we assume that $\Omega$ is a bounded smooth domain of $\mathbb{R}^{n}$ $(n\geq 5)$ and $\lambda\geq0$. For $1\leq p\le\infty$ we denote by $|\cdot|_p$ the $L^p(\Omega)$ norm whereas, we denote by $\|\cdot\|$ the $H^2\cap H_0^1(\Omega)$ norm, that is $\|u\|^2=\int_\Omega|\Delta u|^{2}$. We fix some exponent $q$ with $q>\frac{n% }{4}$ and $q\geq 2$. The definitions and results below do not depend on the special choice of $q$. Consider the functional space \begin{equation*} X(\Omega )=\{ u\in W^{4,q}(\Omega )\mid u=\Delta u=0\text{ on }\partial \Omega \} \,. \end{equation*} Then, we say that a function $u\in L^{2}(\Omega )$, $u\geq 0$ is a \emph{solution} of (\ref{general problem}) if $g(u)\in L^{1}( \Omega)$ and \begin{equation*} \int_{\Omega }u\Delta ^{2}v=\lambda \int_{\Omega }g(u)v\quad \forall v\in X(\Omega). \end{equation*} A solution $u$ of (\ref{general problem}) is called \emph{regular} (resp.\ \emph{singular}) if $u\in L^{\infty }(\Omega)$ (resp.\ $u\notin L^{\infty }( \Omega)$). We also say that a solution $u_{\lambda }$ of (\ref {general problem}) is \emph{minimal} if $u_{\lambda }\leq u$ a.e.\ in $\Omega$ for any further solution $u$ of (\ref{general problem}). Next, we define $$\label{Lambda} \Lambda(g(s)):=\{ \lambda \geq 0:(\ref{general problem})\text{ admits a solution}\}\,,\quad\lambda ^{\ast }(g(s)):=\sup \Lambda(g(s))\,.$$ When it is clear which $g$ we are dealing with we will simply write $\Lambda$ and $\lambda^*$. Clearly, $0\in\Lambda$ so that $\Lambda\neq\emptyset$ and $\lambda^*$ is well-defined. Finally, we call \textbf{extremal} a solution $u^*$ of (\ref{general problem}) with $\lambda =\lambda ^{\ast }$. Our first statement concerns the log-convex case (\ref{log-convex}). We set $f(s):=\log g(s)$, we assume that $$f\in C^{1}(\mathbb{R}_{+})\,,\quad s\mapsto f(s)\text{ is nonconstant increasing and convex} \label{f}$$ so that (\ref{general problem}) reads $$\begin{gathered} \Delta ^{2}u=\lambda e^{f(u)}\quad \text{in }\Omega \\ u=\Delta u=0\quad \text{on }\partial \Omega . \end{gathered} \label{logconvex problem}$$ \begin{theorem} \label{log-convex-theo} Assume that $f$ satisfies \eqref{f}. Then there exists $\lambda ^{\ast }>0$ such that: \begin{itemize} \item[(i)] For $0<\lambda <\lambda ^{\ast }$ problem \eqref{logconvex problem} admits a minimal regular solution. \item[(ii)] For $\lambda =\lambda ^{\ast }$ problem \eqref{logconvex problem} admits at least a solution, not necessarily regular. \item[(iii)] For $\lambda>\lambda^{\ast }$ problem \eqref{logconvex problem} admits no solution. \end{itemize} \end{theorem} Next, we consider the power-type case: $$\begin{gathered} \Delta ^{2}u=\lambda (1+u)^{p}\quad \text{in }\Omega \\ u>0\quad \text{in }\Omega \\ u=\Delta u=0 \text{on }\partial\Omega \,. \end{gathered} \label{p1}$$ Our first result about (\ref{p1}) deals with the subcritical case. In such situation, critical point theory applies. We assume that the minimax variational characterization of mountain pass solutions given by Ambrosetti-Rabinowitz \cite{ar} is known to the reader. \begin{theorem} \label{sottocritic.theo} Assume that $10$ such that: \begin{itemize} \item[(i)] For $0<\lambda <\lambda ^{\ast }$ problem \eqref{p1} admits at least two solutions: the minimal solution and a mountain pass solution. \item[(ii)] For $\lambda =\lambda ^{\ast }$ problem \eqref{p1} admits a unique solution. \item[(iii)] For $\lambda>\lambda^{\ast }$ problem $(\ref{p1})$ admits no solution. \end{itemize} \end{theorem} The supercritical case $p>(n+4)/(n-4)$ is more delicate and we only have partial results. Note that Theorem \ref{log-convex-theo} defines $\lambda^{\ast }(e^{s})>0$. This number is in some sense optimal'' for the following statement: \begin{theorem} \label{supercritic.theo} Assume that $p>(n+4)/(n-4)$. Then there exists $\lambda ^{\ast }\geq \frac{1}{p}\lambda ^{\ast }(e^{s})$ such that: \begin{itemize} \item[(i)] For $0<\lambda <\lambda ^{\ast }$ problem \eqref{p1} admits a minimal solution which is regular whenever $0<\lambda <\frac{1}{p}\lambda ^{\ast }(e^{s})$; \item[(ii)] For $\lambda>\lambda ^{\ast }$ problem \eqref{p1} admits no solutions. \end{itemize} \end{theorem} The upper bound $\frac{1}{p}\lambda ^{\ast }(e^{s})$ for the regularity of minimal solutions is obtained by comparison arguments. Indeed, after a simple transformation, $\frac{1}{p}\lambda ^{\ast }(e^{s})=\lambda ^{\ast }(e^{ps})$ where the optimal'' choice of the function $e^{ps}$ is a consequence of the fact that the function $s\mapsto ps$ is the smallest function $f$ satisfying (\ref{f}) and $e^{f(s)}\geq (1+s)^{p}$. \section{Some analogies between (\ref{secondorder}) and (\ref{general problem})} Throughout this section we deal with general nonlinearities $g$ satisfying $$\label{gggg} \mbox{g\in C^1(\mathbb{R}_{+}) is a nonconstant strictly positive, increasing and convex function.}$$ We collect here some results which will be useful in the sequel. In some cases, we just give some hints of the proofs since they are essentially similar to previous works. In some other cases (especially in Proposition \ref{leasteigenvalue}) we give more details. We first establish some technical lemmas: \begin{lemma} \label{weak.maximum2} For all $w\in L^{1}(\Omega )$ such that $w\geq 0$ a.e.\ in $\Omega$ there exists a unique $u\in L^{1}(\Omega )$ such that $u\geq 0$ a.e.\ in $\Omega$ and which satisfies \begin{equation*} \int_{\Omega }u\Delta ^{2}v=\int_{\Omega }wv \end{equation*} for all $v\in C^{4}(\overline{\Omega})\cap X(\Omega)$. Moreover, there exists $C>0$ (independent of $w$) such that $|u|_1\leq C\left| w\right|_{1}$. \end{lemma} \begin{proof} It is similar to that of \cite[Lemma 1]{brezis} which makes use of a weak form of the maximum principle. This principle is proved in \cite[Lemma 1] {arioli} for polyharmonic equations in the ball under Dirichlet boundary conditions for which one can use Boggio's principle. Under Navier boundary conditions, Boggio's principle is replaced by the (strong) maximum principle for superharmonic functions and general domains $\Omega$ may be chosen. \end{proof} A weak form of the maximum principle reads as follows: \begin{lemma} \label{weak.maximum3} Assume that $u\in L^{1}( \Omega)$ satisfies \begin{equation*} \int_{\Omega }u\Delta ^{2}v\geq 0 \end{equation*} for all $v\in C^{4}(\overline{\Omega })\cap X(\Omega)$ such that $v\geq 0$ in $\Omega$. Then, $u\geq 0$ a.e.\ in $\Omega$. \end{lemma} The proof of this lemma may be obtained using Lemma \ref{weak.maximum2} and arguing as in \cite {arioli,brezis}. From Lemma \ref{weak.maximum3} and arguing as for \cite[Lemma 4]{arioli}, we obtain a weak form of the super-subsolution method: \begin{lemma} \label{soprasol} Assume $(\ref{gggg})$. Let $\lambda>0$, assume that there exists $\overline{u}\in L^{2}( \Omega )$, $\overline{u}\geq 0$ such that $g(\overline{u})\in L^{1}( \Omega )$ and \begin{equation*} \int_{\Omega }\overline{u}\Delta ^{2}v\geq \lambda \int_{\Omega } g(\overline{u})v\quad\forall v\in X(\Omega): v\geq 0\mbox{ a.e. in }\Omega. \end{equation*} Then, there exists a solution $u$ of $(\ref{general problem})$ which satisfies $0\leq u\leq\overline{u}$ a.e.\ in $\Omega$. \end{lemma} By Lemma \ref{soprasol} we infer at once that the set $\Lambda$ defined in (\ref{Lambda}) is an interval. We now show that it is bounded: \begin{lemma} \label{stimelambda*} Assume \eqref{gggg}. Then, $\alpha_g:=\max\{\alpha>0: g(s)\ge\alpha s\ \forall s\ge0\}>0$ and $$\label{implicit} 0<\lambda^*(g(s))<\frac{\lambda _{1}}{\alpha_g}\,,$$ where $\lambda_1$ denotes the first eigenvalue of $\Delta^2$ in $\Omega$ under Navier boundary conditions. \end{lemma} \begin{proof} A standard application of the Implicit Function Theorem implies $\lambda^*>0$. Let $\Phi _{1}$ denote a positive eigenfunction corresponding to $\lambda_1$. Assume that $u\in L^2(\Omega)$ solves (\ref{general problem}), then we have \begin{equation*} \lambda _{1}\int_{\Omega }u\Phi_{1}=\int_{\Omega }u\Delta^{2}\Phi _{1} =\lambda \int_{\Omega }g(u)\Phi_{1}> \lambda\alpha_g\int_{\Omega }u\Phi _{1} \end{equation*} where the last inequality is strict since $g(u)>\alpha_gu$ for small $u$ (recall that $g(0)>0$). The upper bound for $\lambda^*$ now follows at once. \end{proof} We now show that minimal \emph{regular} solutions of \eqref{general problem} are stable. \begin{proposition} \label{leasteigenvalue} Assume $(\ref{gggg})$. Let $\lambda _{0}\in (0,\lambda ^{\ast })$ and suppose that the minimal solution $u_{\lambda _{0}}$ of \eqref{general problem}, with $\lambda =\lambda _{0}$, is regular. Let $\mu _{1}$ denote the least eigenvalue of the linearized operator $\Delta ^{2}-\lambda g'(u_{\lambda _{0}})$ in $u_{\lambda _{0}}$; then $\mu _{1}\geq 0$. Moreover, if there exists $\overline{\lambda }\in (\lambda _{0},\lambda ^{\ast })$ such that also the minimal solution $u_{\overline{\lambda}}$ of $(\ref{general problem})$, with $\lambda =\overline{\lambda}$, is regular, then $\mu _{1}>0$. \end{proposition} \begin{proof} Recall the variational characterization of $\mu _{1}(\lambda)$ for all $\lambda\in(0,\lambda^*)$: \begin{equation*} \mu _{1}(\lambda)=\inf_{w\in H^{2}\cap H_{0}^{1}(\Omega )}\ \frac{\int_{\Omega }|\Delta w|^{2}-\lambda \int_{\Omega }g' (u_{\lambda })w^{2}}{\int_{\Omega }w^{2}}\,. \end{equation*} Clearly, the map $\lambda \mapsto \mu _{1}(\lambda )$ is non increasing and, by Proposition 2 in \cite{arioli}, it is continuous from the left. For contradiction, assume that $\mu_{1}(\lambda _{0})<0$ and define $\widetilde{\lambda }:=\sup \{ \lambda \geq 0:\mu _{1}(\lambda )>0\}$. By the continuity from the left, we have $\mu _{1}(\widetilde{\lambda })\geq 0$ so that $\widetilde{\lambda }<\lambda_0$. If $\mu _{1}(\widetilde{\lambda })>0$, by the second part of Proposition 2 in \cite{arioli}, we get $\mu _{1}(\lambda )>0$ for some $\lambda >\widetilde{\lambda }$, which contradicts the definition of $\widetilde{\lambda }$. So, it must be $\mu _{1}(\widetilde{\lambda })=0$. Fix some $\gamma \in (\widetilde{\lambda },\lambda _{0})$; then, $u_{\gamma }$ is a strict supersolution of (\ref{general problem}) when $\lambda =\widetilde{\lambda }$; but Proposition 3 in \cite{arioli} yields $u_{\widetilde{\lambda }}=u_{\gamma }$ giving again a contradiction. To prove the second statement, assume for contradiction that $\mu_1(\lambda _0)=0$. Taking into account that $\lambda:\mapsto \mu _1(\lambda )$ is non increasing, the just proved first statement yields $\mu_1(\lambda)=0$ for all $\lambda\in[\lambda_0,\overline{\lambda}]$. But then the same argument as before (which uses Proposition 3 in \cite{arioli}) gives a contradiction. \end{proof} Next, we show that the interval $\Lambda$ is closed, provided the minimal solution $u_{\lambda }$ is regular for all $\lambda$ and the nonlinearity $g$ satisfies a growth condition which is verified by (\ref{log-convex}) and (\ref{power.type}). Since by Lemma \ref{soprasol} the map $\lambda \mapsto u_{\lambda }(x)$ is strictly increasing for all $x\in \Omega$, it makes sense to define $$u^{\ast }(x):=\lim_{\lambda \to \lambda ^{\ast }}u_{\lambda }(x)\quad(x\in\Omega)\,. \label{estremal.sol}$$ The following statement tells us that $u^*$ is the \emph{extremal} solution. \begin{proposition} \label{existu*} Assume \eqref{gggg} and $$\lim_{s\to +\infty }\frac{g'(s)s}{g(s)}>1\,. \label{extra-condition}$$ Assume that the minimal solution $u_{\lambda }$ of $(\ref{general problem})$ is regular for all $\lambda \in (0,\lambda ^{\ast })$ and let $u^*$ be as in $(\ref{estremal.sol})$. Then, $u^*\in H^{2}\cap H_{0}^{1}(\Omega )$ and $u^*$ solves $(\ref{general problem})$ for $\lambda =\lambda ^{\ast }$. Moreover, $u_{\lambda }$ $\to u^{\ast }$ in $H^{2}\cap H_{0}^{1}(\Omega )$ as $\lambda \to \lambda ^{\ast }$. \end{proposition} \begin{proof} Let $u_{\lambda }$ be the minimal solution of $(\ref{general problem})$, then: $$\int_{\Omega }u_{\lambda }\Delta ^{2}v=\lambda \int_{\Omega }g(u_{\lambda })v\quad \forall v\in X( \Omega ) , \label{minim.sol}$$ and, by Proposition \ref{leasteigenvalue}, $$\lambda \int_{\Omega }g'(u_{\lambda })u_{\lambda }^{2}\leq \int_{\Omega }( \Delta u_{\lambda }) ^{2}=\lambda \int_{\Omega }g(u_{\lambda })u_{\lambda }. \label{inequality}$$ From (\ref{extra-condition}), it follows that for every $\varepsilon >0$ there exists $C>0$ such that $(1+\varepsilon)g(s)s\leq g'(s)s^{2}+C$ for all $s\geq 0$. Arguing as in \cite{brezis3}, and applying this last inequality and (\ref{inequality}), we get: \begin{equation*} \int_{\Omega }(g'( u_{\lambda }) u_{\lambda }^{2}+C)\geq (1+\varepsilon )\int_{\Omega }g(u_{\lambda })u_{\lambda }\geq (1+\varepsilon )\int_{\Omega }g'(u_{\lambda })u_{\lambda }^{2}, \end{equation*} which gives the existence of a constant $C_{1}>0$ such that: \begin{equation*} \int_{\Omega }g(u_{\lambda })u_{\lambda }0$such that$|u_{\lambda}|_\infty0$such that the solutions$(\lambda ,u)$of \eqref{general problem}, near$(\lambda ^{\ast},u^{\ast })$, form a differentiable curve$t\in (-\delta ,+\delta )\mapsto (\lambda (t),u(t))\in \mathbb{R}_{+}\times C^{4,\alpha } ( \overline{\Omega }) \cap X(\Omega )$, which satisfies:$u(0)=u^{\ast }$,$\lambda (0)=\lambda ^{\ast }$,$\lambda '(0)=0$and$\lambda ''(0)<0$. \end{proposition} \begin{proof} We argue as in \cite{crandall}. Since$\{ u_{\lambda }\} $is bounded in$L^{\infty }(\Omega )$, by elliptic regularity, we deduce the boundedness of$\{ u_{\lambda }\} $also in$W^{4,p}(\Omega )$, for every$p>1$. Then, by Sobolev embedding, we get that, for every$0<\alpha <1$,$\{ u_{\lambda }\} $is bounded in$C^{3,\alpha }(\overline{\Omega })$and so, again by elliptic regularity,$\{u_{\lambda }\}$is also bounded in$C^{4,\alpha }(\overline{\Omega })$. Finally, from the compact embedding$C^{4,\alpha_1}(\overline{\Omega})\subset C^{4,\alpha_2}(\overline{\Omega})$(for every$\alpha_{1}>\alpha_{2}$) we get the claimed convergence. Let us now define the operator$F:( 0,\lambda ^{\ast }] \times C^{4,\alpha }( \overline{\Omega }) \cap X(\Omega )\to C^{0,\alpha }( \overline{\Omega }) $by: \begin{equation*} F(\lambda ,u):=\Delta ^{2}u-\lambda g(u). \end{equation*} It is not difficult to verify that$F(\lambda ,u)$satisfies the hypotheses in \cite[Theorem 3.2]{crandall0}, from which follows the existence of a curve of solutions,$(\lambda (t),u(t))$, such that$u(0)=u^{\ast }$and$\lambda (0)=\lambda ^{\ast }$. To show that$\lambda '(0)=0$and$\lambda ''(0)<0$, it is sufficient to differentiate$t\mapsto F(\lambda (t),u(t))$twice with respect to$t$and evaluate these derivatives at$t=0$. \end{proof} A further step towards a better knowledge of the set of solutions of problem ($\ref{general problem}$) is made by showing that this set is unbounded in$C^{4,\alpha }( \overline{\Omega }) $. Assume (\ref{gggg}) and for every$u\in C^{0,\alpha}(\overline{\Omega })$let$v:=G(\lambda,u)\in C^{0,\alpha }( \overline{\Omega })$be the unique solution of the problem: \begin{equation*} \begin{gathered} \Delta ^{2}v=\lambda g(u)\quad \mbox{in }\Omega \\ v=\Delta v=0\quad \mbox{on }\partial \Omega \,. \end{gathered} \end{equation*} The solutions of (\ref{general problem}) are fixed points of$G$. Furthermore, by elliptic regularity, we have that$v\in C^{4,\alpha }(\overline{\Omega })$and hence, from the compactness of the embedding$C^{4,\alpha}(\overline{\Omega})\subset C^{0,\alpha }(\overline{\Omega })$, we get that$G$is a compact operator from$C^{0,\alpha }( \overline{\Omega }) $into$C^{0,\alpha }( \overline{\Omega }) $. So, if we call$C_{0}$the component of the set \begin{equation*} S:=\{ (\widetilde{\lambda },u)\in ( 0,\lambda ^{\ast }] \times C^{4,\alpha }( \overline{\Omega }) :u\text{ solves } (\ref{general problem})\text{ with }\lambda =\widetilde{\lambda }\} \end{equation*} to which (0,0) belongs, we are in the framework of \cite[Theorem 6.2]{rabinowitz}, from which it follows that: \begin{proposition} Assume$(\ref{gggg})$. Then$C_{0}$is unbounded in$(0,\lambda ^{\ast }]\times C^{4,\alpha}(\overline{\Omega})$. \end{proposition} \section{Some differences between (\ref{secondorder}) and (\ref{general problem}): radial problems} \label{someremarks} In this section we assume that$\Omega=B$(the unit ball). In this case, writing (\ref{general problem}) in its original form of system (\ref{system}% ), by \cite[Theorem 1]{troy} we know that any regular solution of (\ref {general problem}) is radially symmetric and radially decreasing. We discuss separately the exponential case (\ref{logconvex problem}) (when$f(s)\equiv s $) and the power case (\ref{p1}). For the latter, the critical case$p=(n+4)/(n-4)$deserves particular attention. In radial coordinates$r=|x|, (\ref{general problem}) becomes \label{radialgeneral} \begin{aligned} &u^{iv}(r)+\frac{2( n-1) }{r}u'''(r) +\frac{( n-1) ( n-3) }{r^{2}}u''(r)-\frac{( n-1) ( n-3) }{r^{3}}u'(r)\\ &=\lambda g(u(r))\quad r\in [ 0,1) \end{aligned} supported with Navier boundary conditions $$u(1)=u''(1)+(n-1)u'(1)=0\,. \label{boundary1}$$ Moreover, regular solutionsu$are smooth and therefore$r\mapsto u(r)$must be an even function of$r$, namely $$u'(0)=u'''(0)=0\,. \label{regulars}$$ The main purpose of the present section is to highlight several striking differences between (\ref{general problem}) and the corresponding second order problem (\ref{secondorder}). Another purpose of this section is to estimate$\lambda ^{\ast }$. In order to give an upper bound for$\lambda ^{\ast }$we use Lemma \ref{stimelambda*} The estimate (\ref{implicit}) gives $$\lambda ^{\ast }(e^{s})<\frac{\lambda _{1}}{e}\,,\quad \lambda ^{\ast }((1+s)^{p})<\frac{(p-1)^{p-1}}{p^{p}}\lambda _{1}\,. \label{uppersharp}$$ It is well-known that$\lambda _{1}=Z^{4}$, where$Z$is the first zero of the Bessel function$J_{\frac{n-2}{2}}$. According to \cite{as} we have \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline$n$& 5 & 6 & 7 & 8 \\ \hline$\lambda _{1}$& 407.6653 & 695.6191 & 1103.3996 & 1657.0143 \\ \hline \end{tabular} \end{center} To give a lower bound for$\lambda^*$, we seek a supersolution for (\ref{general problem}). For any$C>0$the function $$\label{UC} U_C(r)=C(\frac{2n}{n+3}r^3-3\frac{n+1}{n+3}r^2+1)$$ belongs to$H^2\cap H_0^1(\Omega)$and satisfies the boundary conditions (\ref{boundary1}). We investigate for which$C$and$\lambda$we have$\Delta^2U_C\ge\lambda g(U_C)$. The largest such$\lambda$gives a lower bound for$\lambda^*$. The choice of$U_C$in (\ref{UC}) as a supersolution is probably not optimal. Nevertheless, with Mathematica we could at least optimize the choice of the constant$C$and find the results listed in the tables in the following subsections. The last purpose of this section is to determine the \emph{ghost solutions} as mentioned in the introduction. More precisely, we determine solutions of (\ref{radialgeneral}) satisfying the first boundary condition in (\ref{boundary1}) but \emph{not} the second. Of particular interest is the value of$\lambda_g$corresponding to the ghost solution. We will see that$\lambda_g$may be either larger or smaller than$\lambda^*$; apparently, the former case occurs for subcritical nonlinearities whereas the latter occurs for supercritical nonlinearities. However, this is not a rule, see the case of critical nonlinearities. \subsection{Exponential nonlinearity} When$f(s)=s$, (\ref{logconvex problem}) written in radial coordinates becomes $$\label{exp} u^{iv}(r)+\frac{2(n-1)}{r}u'''(r)+\frac{(n-1)(n-3)} {r^{2}}u''(r) -\frac{(n-1) ( n-3) }{r^{3}} u'(r)= \lambda e^{u(r)}$$$r\in[0,1)$together with the boundary conditions (\ref{boundary1}). As may be checked by a simple calculation, for$\lambda =\lambda _{e}:=8(n-2)(n-4)$the function$U(r):=-4\log r$is a ghost solution, namely it solves (\ref{exp}) and the first boundary condition in (\ref{boundary1}) but not the second boundary condition. Contrary to what happens for the second order equation, the explicit form of a radial singular solution seems not simple to be determined, see also \cite{arioli}. In dimensions$n=5,6,7,8$, the table below shows first for which values of$C $the function$U_C$defined in (\ref{UC}) is a supersolution of (\ref{exp}% ) and the corresponding lower bound for$\lambda^*$. We also give the upper bound obtained with (\ref{uppersharp}). In the fifth column, we quote from \cite{arioli} a lower bound for the extremal value$\lambda^*(D)$of the corresponding \emph{Dirichlet} problem; as for the eigenvalues, it is considerably larger than$\lambda^*$. Finally, in the last column, we quote$\lambda_e$, namely the value of$\lambda$corresponding to the ghost solution: it is considerably smaller than$\lambda^*$. \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline$n$&$C$&$\lambda ^{\ast }\geq $&$\lambda ^{\ast }<$&$\lambda^*(D)\ge$&$\lambda_e\\ \hline 5 & 1.093 & 98.37 & 149.9716 & 235.89 & 24 \\ \hline 6 & 1.132 & 158.48 & 255.9039 & 361.34 & 64 \\ \hline 7 & 1.162 & 234.26 & 405.9180 & 523.16 & 120 \\ \hline 8 & 1.185 & 325.76 & 609.5814 & 724.50 & 192 \\ \hline \end{tabular} \end{center} \subsection{Power-type nonlinearity} In radial coordinates (\ref{p1}) reads \label{e4} \begin{aligned} &u^{iv}(r)+\frac{2(n-1)}{r}u'''(r)+\frac{(n-1)(n-3)} {r^{2}} u''(r)-\frac{(n-1) ( n-3) }{r^{3}}u'(r)\\ &= \lambda (1+u(r))^{p}\quad r\in[0,1) \end{aligned} together with the boundary conditions (\ref{boundary1}). Let us first recall some results for the second order problem corresponding to (\ref{e4}), namely $$\label{e3} -u''(r)-\frac{n-1}{r}u'(r)=\mu(1+u(r))^{p}\,,\quad r\in[0,1)\,.$$ It is well-known \cite{brezis3} that the functionv_p(r)=r^{-2/(p-1)}-1$solves (\ref{e3}) (and satisfies the Dirichlet boundary condition$u(1)=0$) if \begin{equation*} \mu=\mu_p:=\frac{2(np-n-2p)}{(p-1)^2}\,. \end{equation*} Note that$\mu_p>0$if and only if$p>n/(n-2)$; note also that$n/(n-2)$is the critical (largest) trace exponent$q$for which one has$H^1(\Omega)\subset L^{q+1}(\partial\Omega)$. Moreover,$u_p\in H_{0}^{1}(B)$if and only if$p>(n+2)/(n-2)$, the critical Sobolev exponent. For the fourth order problem, we consider the function \begin{equation*} u_p(r)=r^{-4/(p-1)}-1, \end{equation*} which solves (\ref{e4}) if \begin{equation*} \lambda=\lambda_p:=\frac{8(p+1)(2+2p-np+n)(4p-np+n)}{(p-1)^4}\,. \end{equation*} Note that \begin{equation*} \lambda_p>0\ \Longleftrightarrow\ p\in(1,\frac{n+2}{n-2}) \cup(\frac{n}{n-4},\infty) \end{equation*} and that$u_p\in H^{2}\cap H^1_0(B)$if and only if$p>(n+4)/(n-4)$. The number$(n+2)/(n-2)$is the critical exponent for the first order Sobolev inequality while$n/(n-4)$is again the critical trace exponent$q$for the embedding$H^2(\Omega)\subset L^{q+1}(\partial\Omega)$. For$\lambda=\lambda_p$, the function$u_p$is a singular solution of equation (% \ref{e4}) but$u_p$\emph{does not} satisfy the second condition in (\ref {boundary1}); hence, it is not a solution of problem (\ref{general problem}). The functions$u_p$are the ghost solutions. These facts suggest several problems which we quote in Section \ref{problems}. Also for (\ref{e4}) we used the function$U_C$in (\ref{UC}). In dimensions$n=5,6,7,8$, the tables below show both for which values of$C$the function$U_C$is a supersolution of (\ref{e4}) and the corresponding lower bound for$\lambda^*$. We also give the upper bound obtained with (\ref {uppersharp}). The tables correspond, respectively, to the cases$p=3/2$(subcritical) and$p=10$(supercritical); in the first case we have$\lambda^*<\lambda_p$, whereas in the second we have$\lambda^*>\lambda_p$. \begin{center} (p=3/2)\\ \begin{tabular}{|c|c|c|c|c|} \hline$n$&$C$&$\lambda ^{\ast }\geq $&$\lambda ^{\ast }<$&$\lambda_p$\\ \hline 5 & 0.801 & 72.09 & 156.91 & 2800 \\ \hline 6 & 0.844 & 118.16 & \ 267.74 & 1920 \\ \hline 7 & 0.878 & 177 & \ 424.69 & 1200 \\ \hline 8 & 0.905 & 248.79 & 637.78 & 640 \\ \hline \end{tabular} \\[4pt] (p=10)\\ \begin{tabular}{|c|c|c|c|c|} \hline$n$&$C$&$\lambda ^{\ast }\geq $&$\lambda ^{\ast }<$&$\lambda_p$\\ \hline 5 & 0.111 & 9.99 & 15.79 & 1.542 \\ \hline 6 & 0.115 & 16.1 & 26.94 & 6.001 \\ \hline 7 & 0.118 & 23.79 & 42.74 & 12.648 \\ \hline 8 & 1.121 & 33.26 & 64.19 & 21.46 \\ \hline \end{tabular} \end{center} \subsection{The critical case} \label{radcrit} Of special interest is problem (\ref{p1}) in the critical case$p=(n+4)/(n-4)$. By Theorem \ref{sottocritic.theo} and \cite[Theorem 1]{troy} we know that this problem admits at least two regular and radially symmetric solutions. Take any such solution$u$; then, for$\lambda<\lambda^*$, the function$v=\lambda ^{(n-4)/8}(1+u)$solves the problem $$\begin{gathered} \Delta ^{2}v=v^{(n+4)/(n-4)} \quad \text{in }B \\ v>\lambda^{(n-4)/8}\quad \text{in }B \\ v=\lambda^{(n-4)/8}\quad \text{on }\partial B \\ \Delta v=0\quad \text{on }\partial B\,. \end{gathered} \label{critic.pb}$$ Equivalently,$v=v(r)satisfies \label{radialcritical} \begin{aligned} &v^{iv}(r)+\frac{2( n-1) }{r}v'''(r) +\frac{( n-1) ( n-3) }{r^{2}}v''(r)\\ &-\frac{( n-1) ( n-3) }{r^{3}}v'(r)-v(r)^{(n+4)/(n-4)}=0 \end{aligned} with the boundary conditions $$\label{Nbc} v(1)=\lambda^{(n-4)/8}\,,\quad\Delta v(1)=v''(1)+(n-1)v'(1)=0\,,$$ and the regularity conditionsv'(0)=v'''(0)=0$. Consider now the critical problem over the whole space $$\Delta ^{2}v=v^{(n+4)/(n-4)}\quad \text{in }\mathbb{R}^{n}\,. \label{critic.eq}$$ By \cite[Theorem 1.3]{lin}, we know that (up to translations) any smooth positive solution of (\ref{critic.eq}) has the form $$\label{giorgio} v_{d}( x) =\frac{( n( n^{2}-4) ( n-4) d^{2}) ^{(n-4)/8}}{( 1+d\left| x\right| ^{2}) ^{(n-4)/2}}\quad ( d>0) .$$ The main goal of this section is to describe the (smooth) continuation of solutions of (\ref{critic.pb}) \emph{outside}$B$. We obtain a new phenomenon, not available for the corresponding second order problem. \begin{proposition} \label{nonprolung.} Let$v$be a (radial) solution of$(\ref{critic.pb})$; then it does not admit a positive radial extension to$\mathbb{R}^{n}$. \end{proposition} \begin{proof} By contradiction suppose there exists$\overline{v}$, positive radial extension of$v$to$\mathbb{R}^{n}$. Then, by \cite[Theorem 4]{swanson} we have that$\overline{v}$coincides with one of the functions$v_d$in (\ref{giorgio}), for some$d>0$. But this is impossible since for all$d$, the function$v_{d}$does not satisfy the second condition in (\ref{Nbc}). \end{proof} For the critical growth second order problem it is known (see e.g.\ \cite[Theorem 7]{gazzola3}) that the solutions of the equation in fact coincide in$B$with some of the functions$v_d$of the corresponding family (\ref{giorgio}) and it is so clear in which way they are continued. Proposition \ref{nonprolung.} tells us that fourth order problems behave differently: it is therefore natural to inquire in which way the solutions of (\ref{critic.pb}) may be continued for$|x|>1$. To this end, we performed several numerical experiments with Mathematica. The next figures display the graphics of three solutions of $$\label{oden=8} v^{iv}(r)+\frac{14}{r}v'''(r)+\frac{35}{r^{2}} v''(r)-\frac{35}{r^{3}}v'(r)-v(r)^3=0\,.$$ All three solutions satisfy the initial conditions $$\label{firstcond} v(0)=4\sqrt{\frac{6}{5}}\approx4.38178\quad v'(0)=v'''(0)=0.$$ The distinction between the three solutions is made by the choice of the second derivative at$r=0$: we take respectively $$\label{secondcond} \begin{gathered} v''(0)=-\frac{8}{5}\sqrt{\frac{6}{5}}\approx-1.75271\,,\quad v''(0)=-\frac{8}{5}\sqrt{\frac{6}{5}}-10^{-3}\,,\\ v''(0)=-\frac{8}{5}\sqrt{\frac{6}{5}}+10^{-3}\,. \end{gathered}$$ Therefore, the first figure represents the function (\ref{giorgio}) for$n=8$and$d=0.1$. \begin{figure}[hbt] \begin{center} \includegraphics[width=0.5\textwidth]{fig1} \end{center} \caption{The plot of the solution of (\ref{oden=8})-(\ref{firstcond})-(\ref {secondcond})$_1$} \end{figure} \begin{figure}[hbt] \begin{center} \includegraphics[width=0.5\textwidth]{fig2}\\ \includegraphics[width=0.5\textwidth]{fig3} \end{center} \caption{The plots of the solutions of (\ref{oden=8})-(\ref{firstcond}) with (\ref{secondcond})$_2$and (\ref{secondcond})$_3$} \end{figure} We performed further numerical experiments for other choices of$n$and$d$but the results were completely similar. Obviously, if one takes the equilibrium'' initial second derivative (the one of (\ref{giorgio})), then the solution is precisely$v_d$. If one slightly increases this value, the corresponding solution has first a global minimum at positive level and then blows up towards$+\infty$. If one slightly decreases the equilibrium value, the corresponding solution vanishes, becomes negative and then blows up towards$-\infty. These numerical results are partially confirmed by a rigorous proof. To be more precise, up to rescaling we may restrict our attention to the following problem \label{radialcrit} \begin{gathered} \begin{aligned} &u^{iv}(r)+\frac{2(n-1)}{r}u'''(r)+\frac{(n-1)(n-3)}{r^{2}}u''(r)\\ &- \frac{(n-1)(n-3)}{r^{3}}u'(r)=u^{(n+4)/(n-4)}(r)\quad r\in [0,\infty) \end{aligned} \\ u(0)=1\,,\quad u'(0)=u'''(0)=0\,,\quad u''(0)=\gamma<0\,. \end{gathered} Here\gamma$is the only free parameter while$u'(0)=u^{\prime% \prime\prime}(0)=0$are the already mentioned regularity conditions. Existence and uniqueness of a local solution$u_\gamma$of (\ref{radialcrit}) is quite standard. For a suitable choice of$\gamma<0$, say$\gamma=\overline{\gamma}$, the unique solution$\overline{u}:=u_{\overline{\gamma}}$of (\ref{radialcrit}) is in the family (\ref{giorgio}), namely \begin{equation*} \overline{u}(r)=\frac{[n(n^2-4)(n-4)]^{(n-4)/4}} {(\sqrt{n(n^2-4)(n-4)}+r^2)^{(n-4)/2}}\,. \end{equation*} Then we prove the following statement. \begin{theorem} \label{nuovo} For any$\gamma<0$let$u_\gamma$be the unique (local) solution of$(\ref{radialcrit})$. Then: \begin{itemize} \item[(i)] If$\gamma<\overline{\gamma}$there exists$R>0$such that$u_\gamma(R)=0$,$u_\gamma(r)<\overline{u}(r)$and$u'_\gamma(r)<0$for$r\in(0,R]$. \item[(ii)] If$\gamma>\overline{\gamma}$there exist$0\overline{u}(r)$for$r\in(0,R_2)$,$u'_\gamma(r)<0 $for$r\in(0,R_1)$,$u'_\gamma(R_1)=0$,$u'_\gamma(r)>0$for$r\in(R_1,R_2)$and$\displaystyle\lim_{r\to R_2}u_\gamma(r)=+\infty$. \end{itemize} \end{theorem} \begin{remark} \rm The functions$u=u(r)$displayed in the last plot of Figure 2 solve the following Dirichlet problem \begin{gather*} \Delta ^{2}u=u^{(n+4)/(n-4)}\quad \text{in }B_{R} \\ u=\gamma \quad \text{on }\partial B_{R} \\ \frac{\partial u}{\partial \nu }=0\quad \text{on }\partial B_{R} \end{gather*} for some$\gamma ,R>0$. Then, the function$w(x)=\frac{u(Rx)}{\gamma }-1$satisfies \begin{gather*} \Delta ^{2}w=\lambda (1+w)^{(n+4)/(n-4)}\quad \text{in }B \\ w=\frac{\partial w}{\partial \nu }=0\quad \text{on }\partial B \end{gather*} for$\lambda =R^{4}\gamma ^{8/(n-4)}$, namely the Dirichlet problem for the equation in (\ref{p1}) in the unit ball. \end{remark} We conclude this section with the table containing the value of$\lambda_{(n+4)/(n-4)}$and the estimates of$\lambda^*$obtained with$U_C$in (\ref{UC}): \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline$n$&$(n+4)/(n-4)$&$\lambda _{(n+4)/(n-4)}$&$C$&$\lambda ^{\ast }\geq $&$\lambda ^{\ast }<$\\ \hline 5 & 9 & 25/16 & 0.123 & 11.07 & 17.65 \\ \hline 6 & 5 & 9 & 0.235 & 32.9 & 56.98 \\ \hline 7 & 11/3 & 441/16 & 0.335 & 67.54 & 128.72 \\ \hline 8 & 3 & 64 & 0.425 & 116.84 & 245.48 \\ \hline \end{tabular} \end{center} \section{Proof of Theorem \ref{log-convex-theo}} Note first that, up to rescaling$\lambda$, we may assume that$f(0)=0$. Then, we start with a calculus'' statement. \begin{lemma} \label{convex} Assume that$\varphi \in C^{1}[ 0,+\infty ) $is a nonnegative, non-decreasing and convex function such that$\varphi (0)=0$. Then for any$x\geq 0$and any$\beta >1$we have$\varphi (\beta x)\geq \beta \varphi (x)$. \end{lemma} \begin{proof} It follows at once from the inequality$\frac{d}{dx}(\varphi(\beta x)-\beta\varphi(x))\ge0$. \end{proof} We now establish an improved version of \cite[Lemma 5]{arioli}: \begin{lemma} \label{lemma.log.exp} Assume that for some$\mu >0$there exists a (possibly singular) solution$u_{0}$of$(\ref{logconvex problem})$with$\lambda =\mu$. Then, for all$0<\lambda<\mu$there exists a regular solution of$(\ref {logconvex problem})$. \end{lemma} \begin{proof} Let$0<\lambda <\mu $, and consider the (unique) functions$u_1,u_2\in L^2(\Omega )$satisfying respectively \begin{equation*} \int_{\Omega }u_{1}\Delta ^{2}v=\lambda \int_{\Omega }e^{f(u_{0})}v\quad \mbox{and} \quad\int_{\Omega }u_{2}\Delta ^{2}v=\lambda \int_{\Omega }e^{f(u_{1})}v\quad \forall v\in X(\Omega )\,; \end{equation*} such functions exist by Lemma \ref{weak.maximum2} and belong to$L^{2}(\Omega )$since Lemma \ref{weak.maximum3} entails \begin{equation*} u_0\ge\frac{\lambda}{\mu}u_0=u_1\geq u_2 \quad \text{a.e. in }\Omega \,. \end{equation*} We now need the following elementary statement: For all$\vartheta >1$and all$\alpha >1$, there exists$\gamma >0$such that $$\label{elementary} e^{\vartheta f(s)}+\gamma -\alpha e^{f(s)}\geq 0 \quad \forall s\geq 0.$$ Fix$\vartheta:=\mu/\lambda >1$and take$\alpha> \max\{\frac{n}{4},2\}$; then, (\ref{elementary}) ensures that there exists$k>0$such that $$\label{elementary2} e^{\frac{\mu}{\lambda}f(s)}+\frac{k}{\lambda }\geq \alpha e^{f(s)} \quad\forall s\geq 0.$$ Let$w\in X(\Omega)$be the unique solution of the equation$\Delta^{2}w =k$in$\Omega$; then$w\in L^\infty(\Omega)$and$w>0$in$\Omega. Moreover, using Lemma \ref{convex} and (\ref{elementary2}) we get \begin{align*} \int_{\Omega }(u_{1}+w)\Delta ^{2}v &=\lambda \int_{\Omega }\big(e^{f(u_{0})}+\frac{k}{\lambda }\big) v\\ &= \lambda \int_{\Omega }\big( e^{f(\frac{\mu }{\lambda }u_{1})} +\frac{k}{\lambda }\big) v\\ &\geq \lambda \int_{\Omega }\big( e^{\frac{\mu }{\lambda }f(u_{1})} +\frac{k}{\lambda }\big) v \\ &\geq \lambda \alpha \int_{\Omega }e^{f(u_{1})}v\\ &=\alpha\int_{\Omega }u_{2}\Delta ^{2}v \end{align*} for allv\in X(\Omega)$such that$v\geq 0$in$\Omega$. Hence, by Lemma \ref{weak.maximum3}, we infer that$u_{2}\leq \frac{u_{1}+w}{\alpha }$. Since$\alpha>2$and$w>0$, this inequality, together with the monotonicity and convexity of$f$, implies that \begin{equation*} f(u_{2})\leq f( \frac{u_{1}}{\alpha }+\frac{w}{\alpha }) \leq f\big( \frac{1}{\alpha }u_{1}+( 1-\frac{1}{\alpha }) w \big) \leq \frac{1}{\alpha }f(u_{1})+( 1-\frac{1}{\alpha }) f(w). \end{equation*} In particular, \begin{equation*} e^{f(u_{2})}\leq e^{\frac{1}{\alpha }f(u_{1})}e^{( 1-\frac{1}{\alpha }% ) f(w)}\ ; \end{equation*} since$e^{\frac{1}{\alpha }f(u_{1})}\in L^{\alpha }(\Omega )$and$e^{(1-\frac{1}{\alpha }) f(w)}\in L^{\infty }(\Omega )$we get at once that$e^{f(u_{2})}\in L^{\alpha }( \Omega )$. Finally, consider$u_{3}\in L^{2}( \Omega ) $such that \begin{equation*} \int_{\Omega }u_{3}\Delta ^{2}v=\lambda \int_{\Omega }e^{f(u_{2})}v \quad \forall v\in X(\Omega )\,. \end{equation*} By elliptic regularity and the fact that$\alpha >\frac{n}{4}$, we deduce$u_{3}\in W^{4,\alpha }( \Omega ) \subset L^{\infty }( \Omega)$. Moreover, \begin{equation*} \int_\Omega(u_2-u_3)\Delta^2v=\lambda\int_\Omega(e^{f(u_1)}- e^{f(u_2)})v\ge0\quad\forall v\in X(\Omega): v\ge0 \text{ in } \Omega \end{equation*} so that by Lemma \ref{weak.maximum3} we infer that$u_{3}\leq u_{2}$. Hence, \begin{equation*} \int_{\Omega }u_{3}\Delta ^{2}v\geq \lambda \int_{\Omega }e^{f(u_{3})}v\text{ \ \ \ }\forall v\in X(\Omega): v\geq 0\text{ in }\Omega . \end{equation*} Then$u_{3}$is a weak bounded supersolution of (\ref{logconvex problem}) and the statement follows by Lemma \ref{soprasol}. \end{proof} Theorem \ref{log-convex-theo} is now a straightforward consequence of (\ref{implicit}) and of Lemma \ref{lemma.log.exp} and Proposition \ref{existu*}. \section{Proof of Theorem \ref{sottocritic.theo}} \label{dimteosottocrit} The proof is obtained by combining some well-known results in \cite{ar,brezis2,crandall,ge,van der vorst}. Firstly, by applying the regularity results in \cite{van der vorst}, we prove the following statement. \begin{proposition} \label{reg.prop} Assume that$10$there exist$q_{\varepsilon }\in L^{\frac{n}{4}}( \Omega ) $and$F_{\varepsilon }\in L^{\infty }( \Omega )$such that: $$(1+u(x))^{p}=q_{\varepsilon }(x)u(x)+F_{\varepsilon }(x)\quad\mbox{and}\quad \|q_{\varepsilon }\| _{\frac{n}{4}}<\varepsilon\,. \label{f3}$$ Fix$M\geq 1$and write $$(1+u)^{p}=\chi _{\left\{ u\leq M\right\} }(1+u)^{p}+\chi _{\left\{ u>M\right\} }(1+u)^{p}=\varphi (x)+\chi _{\left\{ u>M\right\} } \frac{(1+u)^{p}}{u}u \label{f2}$$ where$\chi _{\{.\} }$is the characteristic function and$\varphi (x)=\chi _{\{ u\leq M\} }(1+u)^{p}\in L^{\infty }(\Omega )$. It is clear that$(1+u)^{p}\le(2u)^{p}$whenever$u>M$. Moreover, using the embedding$H^{2}(\Omega )\subset L^{2n/(n-4)}(\Omega )$and the fact that$p\leq (n+4)/(n-4)$, we have that$u^{p-1}\in L^{\frac{n}{4}}( \Omega ) $, hence: \begin{equation*} 0\le a(x):=\chi _{\left\{ u>M\right\} }\frac{(1+u)^{p}}{u}\leq 2^{p}u^{p-1}\in L^{\frac{n}{4}}( \Omega ) . \end{equation*} Therefore, we may write$(1+u)^p=\varphi(x)+a(x)u$with$\varphi\in L^\infty(\Omega)$and$a\in L^{\frac{n}{4}}(\Omega)$. Applying \cite[Lemma B2]{van der vorst}, for every$\varepsilon>0$we obtain $$a(x)u(x)=q_{\varepsilon }(x)u(x)+f_{\varepsilon }(x) \label{f1}$$ where$q_{\varepsilon }$and$f_{\varepsilon }$satisfy$\|q_{\varepsilon }\| _{\frac{n}{4}}<\varepsilon$and$f_{\varepsilon}\in L^\infty(\Omega)$. Defining$F_{\varepsilon }(x)=f_{\varepsilon }(x)+\varphi (x)$, from (\ref{f2}) and (\ref{f1}) we obtain (\ref{f3}). By (\ref{f3}), for every$\varepsilon >0$, the equation in (\ref{p1}) can be rewritten as \begin{equation*} \Delta ^{2}u=\lambda\big(q_{\varepsilon }(x)u(x)+F_{\varepsilon }(x)\big) \quad \text{in }\Omega \end{equation*} so that the result follows by Steps 2 and 3 in \cite{van der vorst}. \end{proof} Consider the functional \begin{equation*} J(u)=\int_{\Omega }\left| \Delta u\right| ^{2}-\frac{\lambda }{p+1}% \int_{\Omega }|1+u|^{p+1}. \end{equation*} When$10$in$\Omega $so that$v$solves the problem \begin{gather*} \Delta ^{2}v=\lambda(1+u_{\lambda }+v)^{(n+4)/(n-4)}-\lambda (1+u_{\lambda })^{(n+4)/(n-4)}\quad \text{in }\Omega \\ v>0\quad \text{in }\Omega \\ v=\Delta v=0\quad \text{on }\partial \Omega . \end{gather*} Setting$h(x,v)=\lambda (1+u_{\lambda }+v)^{(n+4)/(n-4)}-\lambda (1+u_{\lambda })^{(n+4)/(n-4)}-\lambda v^{(n+4)/(n-4)}$, the previous problem reads \begin{gather*} \Delta ^{2}v=\lambda v^{(n+4)/(n-4)}+h(x,v)\quad \text{in }\Omega \\ v>0\quad \text{in }\Omega \\ v=\Delta v=0\quad \text{on }\partial \Omega \end{gather*} Finally, let$w=\lambda ^{(n-4)/8}v$and$f(x,w)=\lambda ^{(n-4)/8}h(x,\lambda ^{(4-n)/8}w)$, then$w$satisfies $$\begin{gathered} \Delta ^{2}w=w^{(n+4)/(n-4)}+f(x,w)\quad \text{in }\Omega \\ w>0\quad \text{in }\Omega \\ w=\Delta w=0\quad \text{on }\partial \Omega \end{gathered} \label{p3}$$ The function$f$satisfies the hypotheses in \cite[Corollary 1]{ge}; therefore, we infer the existence of a positive solution of (\ref{p3}) or, equivalently, of a positive mountain pass solution for (\ref{p1}). To conclude the proof of Theorem \ref{sottocritic.theo}, we need to show that the extremal solution$u^{\ast }$, which exists by Proposition \ref{existu*}, is unique. To this end, recall that$u^{\ast }$is a classical solution in view of Proposition \ref{reg.prop}. Therefore, it suffices to argue as for Lemma 2.6 in \cite{brezis3}. \section{Proof of Theorem \ref{supercritic.theo}} As we have already observed,$\lambda^{\ast }(e^{ps})=\frac{1}{p}\lambda ^{\ast }(e^{s})$then, since$e^{ps}\geq (1+s)^{p}$for all$s\geq 0$, arguing as in \cite[Theorem 8]{gazzola3}, we obtain $$0<\frac{1}{p}\lambda ^{\ast }(e^{s})\leq \lambda ^{\ast }((1+s)^{p}). \label{monoton.param.}$$ By Lemma \ref{lemma.log.exp}, for every$\lambda <\frac{1}{p}\lambda ^{\ast }(e^{s})$there exists a minimal regular solution$u_{\lambda }$of (\ref {logconvex problem}) with$f(s)=ps$. Such$u_{\lambda }$is also a bounded supersolution of (\ref{p1}), indeed \begin{equation*} \int_{\Omega }u_{\lambda }\Delta ^{2}v=\lambda \int_{\Omega }e^{pu_{\lambda }}v\geq \lambda \int_{\Omega }(1+u_{\lambda })^{p}v\quad \forall v\in X(\Omega ):v\geq 0\text{ in }\Omega . \end{equation*} Then, by Lemma \ref{soprasol}, for all$\lambda <\frac{1}{p}\lambda ^{\ast}(e^{s})$there exists a solution$u_{p}$of (\ref{p1}) such that$u_{p}\leq u_{\lambda }$. \section{Proof of Theorem \ref{nuovo}}$(i)$Since$u_\gamma(0)=\overline{u}(0)$,$u_\gamma'(0)=\overline{u}'(0)$and$u_\gamma''(0)<\overline{u}''(0)$, we have$u_\gamma(r)<\overline{u}(r)$at least in a sufficiently small right neighborhood of$r=0$. For contradiction, assume that there exists (a first)$\rho>0$such that $$\label{R} u_\gamma(\rho)=\overline{u}(\rho)\,,\quad u_\gamma(r)<\overline{u} (r)<1\quad\forall r\in(0,\rho)\,.$$ Note that (\ref{radialcrit}) may be rewritten as $$\label{first} \left\{r^{n-1}\left[\Delta u_\gamma(r)\right]'\right\}'= r^{n-1}u_\gamma^{(n+4)/(n-4)}(r)\,,\quad \left\{r^{n-1}\left[\Delta% \overline{u}(r)\right]'\right\}'= r^{n-1}\overline{u}^{\frac{% n+4}{n-4}}(r)$$ for all$r\in[0,\rho]$. By subtracting the two equations in (\ref{first}) we readily obtain $$\label{difference} \{r^{n-1}\left[\Delta u_\gamma(r)-\Delta\overline{u}(r)\right] '\}'=r^{n-1} [u_\gamma^{(n+4)/(n-4)}(r)-\overline{u% }^{(n+4)/(n-4)}(r)]\quad\forall r\in[0,\rho]\,.$$ Since both solutions$u_\gamma$and$\overline{u}$are smooth, we have \begin{equation*} \lim_{r\to0}\left\{r^{n-1} \left[\Delta u_\gamma(r)-\Delta\overline{u}(r)% \right]'\right\}=0\,; \end{equation*} therefore, for any$r\in(0,\rho]$we may integrate (\ref{difference}) over$[0,r]$and obtain $$\label{integrate} r^{n-1}[\Delta u_\gamma(r)-\Delta\overline{u}(r)] '=\int_0^r t^{n-1}[u_\gamma^{(n+4)/(n-4)}(t) -\overline{u}^{(n+4)/(n-4)}(t)]\, dt <0$$ for all$r\in(0,\rho]$, the last inequality being a consequence of (\ref{R}). Note also that$\Delta u_\gamma(0)=n\gamma0\quad\mbox{in }B_\rho\,. Moreover, (\ref{R}) tells us that $(u_\gamma-\overline{u})=0$ on $\partial B_\rho$. This, together with (\ref{super}) and the maximum principle shows that $u_\gamma>\overline{u}$ in $B_\rho$. This contradicts (\ref{R}) and shows that $u_\gamma(r)<\overline{u}(r)$ as long as $u_\gamma(r)$ remains positive. The positivity interval for $u_\gamma$ cannot be $(0,\infty)$, otherwise $u_\gamma$ would be a positive solution of (\ref{critic.eq}) which is not in the family (\ref{giorgio}), against \cite[Theorem 1.3]{lin}. We have so far proved that there exists a finite $R>0$ such that $u_\gamma(R)=0$ and $u_\gamma(r)<\overline{u}(r)$ whenever $r\in(0,R]$. We now show that $u'_\gamma(r)<0$ for all $r\in(0,R]$. If $u_\gamma'(R_\gamma)=0$ for some $R_\gamma\le R$, then $\Delta u_\gamma(R_\gamma)\ge0$; by integrating the first of (\ref{first}) over $[0,r]$ for $r>R_\gamma$ and arguing as above we deduce that $\Delta u_\gamma(r)>0$ for all $r>R_\gamma$ and, in turn, that $u_\gamma^{% \prime}(r)>0$ for all $r>R_\gamma$. But then we would find $\rho>R_\gamma$ such that (\ref{R}) holds, which we have just seen to be impossible. This contradiction shows that $u'_\gamma(r)<0$ for all $r\in(0,R]$ and completes the proof of $(i)$. \smallskip \noindent $(ii)$ As in the proof of $(i)$, it cannot be $u_\gamma(\rho)=\overline{u}(\rho)$ for some $\rho>0$. Hence, for $r>0$, $0<\overline{u}(r)0$ such that $u'_\gamma(R_1)=0$, then $u'_\gamma(r)<0$ for all $r>0$ so that $u_\gamma$ would be a positive global solution of (\ref{critic.eq}) which is not in the family (\ref{giorgio}), against \cite[Theorem 1.3]{lin}. So, let $R_1>0$ be the first solution of $u_\gamma'(R_1)=0$; then, $\Delta u_\gamma(R_1)\ge0$. By integrating the first of (\ref{first}) over $[0,r]$ for $r>R_1$ we deduce that $\Delta u_\gamma(r)>0$ for all $r>R_1$ and that $u_\gamma'(r)>0$ for all $r>R_1$. Invoking once more \cite[Theorem 1.3]{lin}, we deduce that $u_\gamma$ cannot exist globally; this proves the existence of $R_2$ and completes the proof of $(ii)$. \section{Some unsolved problems} \label{problems} \begin{problem} \label{limitatezza} Prove Lemma \ref{lemma.log.exp} for general nonlinearities $g$. \end{problem} For any strictly positive, increasing and convex function $g$, it is shown in \cite{brezis} that (\ref{secondorder}) possesses a minimal \emph{regular} solution for all $\mu<\mu^*$ (the extremal value). The proof takes advantage of the inequality $\Delta\Phi(u)\le\Phi'(u)\Delta u$ which holds for any smooth concave function $\Phi$ with bounded first derivative and such that $\Phi(0)=0$. For the fourth order problem (\ref{general problem}), this inequality seems out of reach and one should find other issues. On the other hand, the method used in Lemma \ref{lemma.log.exp} seems to apply only to functions $g$ satisfying (\ref{log-convex}). \begin{problem} \label{criticaldimension} Find the critical dimensions. \end{problem} Consider again the second order equation (\ref{secondorder}). For $g(s)=e^s$, it is proved in \cite[Th\'eor\eme 3]{mignot} that if $n\le9$ then $u^*$ is bounded, whereas from \cite{brezis3} we know that if $n\ge10$ and $\Omega$ is a ball, then $u^*$ is unbounded. We call critical dimension $N(g(s))$ the largest dimension for which the semilinear equation with nonlinearity $g$ admits a regular extremal solution in any domain $\Omega$. Then, we just saw that for second order equations we have $N(e^s)=9$. One is then interested in finding the critical dimensions also for fourth order problems. Two main difficulties arise. First, the counterpart of \cite {brezis3} fails due to the double boundary condition and no interpretation in terms of remainder terms for Hardy inequality is available, see \cite{ggm}. Second, also the method in \cite{mignot} fails since the very same arguments as in the proof of \cite[Th\'eor\eme 3]{mignot} yield \begin{equation*} \frac{\lambda a}{4}\int_\Omega[e^{(a+1)u_\lambda}-e^{u_\lambda}] + \frac{a^4}{16}\int_\Omega[e^{au_\lambda}|\nabla u_\lambda|^4] \ge \lambda\int_\Omega[e^{(a+1)u_\lambda}-2e^{(a+2)u_\lambda/2} +e^{u_\lambda}] \end{equation*} for all $a>0$ which allows no conclusion. If one assumes (with no motivation!) that the additional term $\int e^{au_\lambda}|\nabla u_\lambda|^4$ is a lower order term as $\lambda\to\lambda^*$, then we would have boundedness of the extremal solution for $n<20$. Nevertheless, as in \cite{arioli}, we believe that $N(e^s)=12$ for fourth order problems and that the critical dimension does not depend on the boundary condition (Navier or Dirichlet) considered. For the critical dimensions when $g(s)=(1+s)^p$, we refer to \cite[Th\'eor\eme 4]{mignot} and \cite{brezis3}. \begin{problem} \label{uniqureichel} Prove uniqueness for small $\lambda$. \end{problem} If $\Omega$ is conformally contractible, then Reichel \cite{reichel} proves that the equation in (\ref{general problem}) under Dirichlet boundary conditions admits a unique smooth solution for small $\lambda$ and suitable nonlinearities $g$. Conformally contractible domains are slightly more general than starshaped domains and allow to obtain uniqueness from a strict variational principle by means of a Pohozaev-type identity. Among other tools, the proof is based on a crucial extension argument (see Proposition 8 p.68 in \cite{reichel}) which is not available under Navier boundary conditions. Is it possible to by-pass this difficulty and to obtain uniqueness for small $\lambda$ also under Navier boundary conditions? \begin{problem}\label{nodalradial} Nonexistence of entire nodal radial solutions of the critical growth equation. \end{problem} The numerical results of Section \ref{radcrit} and Theorem \ref{nuovo} suggest the following conjecture: the equation $$\label{tttt} \Delta^2u=|u|^{8/(n-4)}u\quad\mbox{in }\mathbb{R}^n$$ admits no radial sign changing solutions. Even if this result is well-known for the second order equation $-\Delta u=|u|^{4/(n-2)}u$, this conjecture appears hard to prove due to a lack of Lyapunov functional for (\ref {radialcritical}). 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