\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 102, pp. 1--9.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/102\hfil Existence of positive solutions] {Existence of positive solutions for higher order singular sublinear elliptic equations} \author[I. Bachar\hfil EJDE-2006/102\hfilneg] {Imed Bachar} \address{Imed Bachar \newline D\'{e}partement de math\'{e}matiques, Facult\'{e} des sciences de Tunis, campus universitaire, 2092 Tunis, Tunisia} \email{Imed.Bachar@ipeit.rnu.tn} \date{} \thanks{Submitted May 10, 2006. Published August 31, 2006.} \subjclass[2000]{34B27, 35J40} \keywords{Green function; higher-order elliptic equations; \hfill\break\indent positive solution; Schauder fixed point theorem} \begin{abstract} We present existence result for the polyharmonic nonlinear problem \begin{gather*} (-\Delta )^{pm} u=\varphi (.,u)+\psi (.,u),\quad \text{in }B \\ u>0,\quad \text{in }B \\ \lim_{|x|\to 1} \frac{(-\Delta )^{jm}u(x)}{(1-|x|)^{m-1}}=0, \quad 0\leq j\leq p-1, \end{gather*} in the sense of distributions. Here $m,p$ are positive integers, $B$ is the unit ball in $\mathbb{R}^{n}(n\geq 2)$ and the nonlinearity is a sum of a singular and sublinear terms satisfying some appropriate conditions related to a polyharmonic Kato class of functions $\mathcal{J}_{m,n}^{(p)}$. \end{abstract} \maketitle \numberwithin{equation}{section} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{Introduction} In this paper, we investigate the existence and the asymptotic behavior of positive solutions for the following iterated polyharmonic problem involving a singular and sublinear terms: \begin{equation} \begin{gathered} (-\Delta )^{pm} u=\varphi (.,u)+\psi (.,u),\quad\text{in }B \\ u>0\quad \text{in }B \\ \lim_{|x|\to 1} \frac{(-\Delta )^{jm}u(x)}{ (1-|x|)^{m-1}}=0, \quad\text{for }0\leq j\leq p-1, \end{gathered} \label{1.1} \end{equation} in the sense of distributions. Here $B$ is the unit ball of $\mathbb{R}^{n}$ $(n\geq 2)$ and $m,p $ are positive integers. This research is a follow up to the work done by Shi and Yao \cite{SY}, who considered the problem \begin{equation} \begin{gathered} \Delta u+k(x)u^{-\gamma }+\lambda u^{\alpha }=0, \quad \text{in }D, \\ u>0, \quad \text{in }D \end{gathered} \label{1.2} \end{equation} where $D$ is a bounded $C^{1,1}$ domain in $\mathbb{R}^{n}(n\geq 2)$, $\gamma ,\alpha $ are two constants in $(0,1),\lambda $ is a real parameter and $k$ is a H\"{o}lder continuous function in $\overline{\Omega }$. They proved the existence of positive solutions. Choi, Lazer and Mckenna in \cite{CM} and \cite{LM} have studied a variety of singular boundary value problems of the type $\Delta u+p(x)u^{-\gamma }$, in a regular domain $D$, $u=0$ on $\partial D$, where $\gamma >0$ and $p$ is a nonnegative function. They proved the existence of positive solutions. This has been extended by M\^{a}agli and Zribi \cite{MZ} to the problem $\Delta u=-f(.,u)$ in $D$, $u=0$ on $\partial D$, where $f(x,.)$ is nonnegative and nonincreasing on $(0,\infty)$. On the other hand, problem \eqref{1.1} with a sublinear term $\psi (.,u)$ and a singular term $\varphi (.,u)=0$, has been studied by M\^{a}agli, Toumi and Zribi in \cite{MTZ} for $p=1$ and by Bachar \cite{B} for $p\geq 1$. Thus a natural question to ask, is for more general singular and sublinear terms combined in the nonlinearity, whether or not the problem \eqref{1.1} has a solution, which we aim to study in this paper. Our tools are based essentially on some inequalities satisfied by the Green function $\Gamma _{m,n}^{(p)}$ (see \eqref{2.1} below) of the polyharmonic operator $u\mapsto (-\Delta )^{pm}u$, on the unit ball $B $ of $\mathbb{R}^{n}$ $(n\geq 2)$ with boundary conditions $( \frac{\partial }{\partial \nu }) ^{j}(-\Delta )^{im}u\big|_{{\partial B}}=0$, for $0\leq i\leq p-1$ and $0\leq j\leq m-1$, where $\frac{\partial }{\partial \nu }$ is the outward normal derivative. Also, we use some properties of functions belonging to the polyharmonic Kato class $\mathcal{J}_{m,n}^{(p)}$ which is defined as follows. \begin{definition}[\cite{B}] \label{def1.1}\rm A Borel measurable function $q$ in $B$ belongs to the class $\mathcal{J}_{m,n}^{(p)}$ if $q$ satisfies the condition \begin{equation} \lim_{\alpha \to 0}\Big(\sup_{x\in B}\int_{B\cap B(x,\alpha )}(\frac{\delta (y)}{\delta (x)})^{m}\Gamma _{m,n}^{(p)}(x,y)|q(y)|dy\Big)=0, \label{1.3} \end{equation} where $\delta (x)=1-|x|$, denotes the Euclidean distance between $x$ and $\partial B$. \end{definition} Typical examples of elements in the class $\mathcal{J}_{m,n}^{(p)} $ are functions in $L^{s}(B)$, with \[ s>\frac{n}{2pm} \quad \text{if } n>2pm \] or with \[ s>\frac{n}{2(p-1)m}, \quad\text{if }2(p-1)mn$ and we assume that the functions $\varphi $ and $\psi $ satisfy the following hypotheses: \begin{itemize} \item[(H1)] $\varphi $ is a nonnegative Borel measurable function on $B\times (0,\infty )$, continuous and nonincreasing with respect to the second variable. \item[(H2)] For each $c>0$, the function $x\mapsto \varphi (x,c(\delta(x))^{m}) / (\delta (x))^{m}$ is in $\mathcal{J}_{m,n}^{(1)}$. \item[(H3)] For each $c>0$, the function $x\mapsto \varphi ( x,c( \delta ( x)) ^{m}) $ is in $L^{r}(B)$. \item[(H4)] $\psi $ is a nonnegative Borel measurable function on $B\times [0,\infty )$, continuous with respect to the second variable such that there exist a nontrivial nonnegative function $h\in L_{\rm loc}^{1}(B)$ and a nontrivial nonnegative function $k\in \mathcal{J}_{m,n}^{(1)}$ such that \begin{equation} h(x)f(t)\leq \psi (x,t)\leq (\delta (x))^{m}k(x)g(t),\quad \text{for }(x,t)\in B\times (0,\infty ), \label{1.4} \end{equation} where $f:[0,\infty )\to [0,\infty )$ is a measurable nondecreasing function satisfying \begin{equation} \lim_{t\to 0^{+}}\frac{f(t)}{t}=+\infty , \label{1.5} \end{equation} and $g$ is a nonnegative measurable function locally bounded on $[0,\infty )$ satisfying \begin{equation} \limsup_{t\to \infty } \frac{g(t)}{t}<\|V_{p} ((\delta (.))^{m}k)\| _{\infty }. \label{1.6} \end{equation} \item[(H5)] The function $x\mapsto (\delta (x))^{m}k(x)$ is in $L^{r}(B)$. \end{itemize} Using a fixed point argument, we shall prove the following existence result. \begin{theorem} \label{thm1.1} Assume (H1)--(H5). Then \eqref{1.1} has at least one positive solution $u\in C^{2pm-1}(B)$, such that $$ a_{j}(\delta (x))^{m}\leq (-\Delta )^{jm}u(x)\leq V_{p-j}(\varphi (.,a_{j}(\delta (.))^{m}))(x)+b_{j}V_{p-j}((\delta (.))^{m}k) (x), $$ for $j\in \{0,\dots ,p-1\} $. In particular, \[ a_{j}(\delta (x))^{m}\leq (-\Delta )^{jm}u(x)\leq c_{j}(\delta (x))^{m}, \] where $a_{j},b_{j},c_{j}$ are positive constants. \end{theorem} Typical examples of nonlinearities satisfying (H1)--(H5) are: \[ \varphi (x,t)=k(x)(\delta (x))^{m\gamma +m}t^{-\gamma }, \] for $\gamma \geq 0$, and \[ \psi (x,t)=k(x)(\delta (x))^{m}t^{\alpha }Log(1+t^{\beta }), \] for $\alpha ,\beta \geq 0$ such that $\alpha +\beta <1$, where $k$ is a nontrivial nonnegative functions in $L^{r}(B)$. Recently Ben Othman \cite{Be} considered \eqref{1.1} when $p=1 $ and the functions $\varphi ,\psi $ satisfy hypotheses similar to the ones stated above. Then she proved that \eqref{1.1} has a positive continuous solutions $u$ satisfying \[ a_{0}(\delta (x))^{m}\leq u(x)\leq V_{1}(\varphi (.,a_{0}(\delta (.))^{m}))(x)+b_{0}V_{1}((\delta (.))^{m-1}k)(x). \] Here we prove an existence result for the more general problem \eqref{1.1} and obtain estimates both on the solution $u$ and their derivatives $(-\Delta )^{jm}u$, for all $j\in \{1,\dots ,p-1\}$. To simplify our statements, we define some convenient notations: \begin{itemize} \item[(i)] Let $B=\{x\in \mathbb{R}^{n}:|x|<1\}$ and let $\overline{B}=\{x\in \mathbb{R}^{n}:|x|\leq 1\}$, for $n\geq 2$. \item[(ii)] $\mathcal{B}(B)$ denotes the set of Borel measurable functions in $B$, and $\mathcal{B}^{+}(B)$ the set of nonnegative ones. \item[(iii)] $C(\overline{B})$ is the set of continuous functions in $\overline{B}$. \item[(iv)] $C^{j}(B)$ is the set of functions having derivatives of order $\leq j$, continuous in $B$ $(j\in \mathbb{N})$. \item[(v)] For $x,y\in B$, $[x,y]^{2}=|{x-y}|^{2}+(1-|x| ^{2})(1-|y|^{2})$. \item[(vi)] Let $f$ and $g$ be two positive functions on a set $S$. We call $f\preceq g$, if there is $c>0$ such that $ f(x)\leq cg(x)$, for all $x\in S$. \\ We call $f\sim g$, if there is $c>0$ such that $\frac{1}{c}g(x)\leq f(x)\leq cg(x)$, for all $x\in S$. \item[(vii)] For any $q\in \mathcal{B}(B)$, we put \[ \|q\|_{m,n,p}:=\sup_{x\in B}\int_{B}( \frac{\delta (y)}{\delta (x)})^{m}\Gamma _{m,n}^{(p)}(x,y)| q(y)|dy. \] \end{itemize} \section{Properties of the iterated Green function and the Kato class } Let $m\geq 1$, $p\geq 1$ be a positive integer and $\Gamma _{m,n}^{(p)}$ be the iterated Green function of the polyharmonic operator $u\mapsto (-\Delta )^{pm}u$, on the unit ball $B$ of $\mathbb{R}^{n}$ $(n\geq 2)$ with boundary conditions $(\frac{\partial }{\partial \nu }) ^{j}(-\Delta )^{im}u\big|_{{\partial B}}=0$, for $0\leq i\leq p-1$ and $0\leq j\leq m-1$, where $\frac{\partial }{\partial \nu }$ is the outward normal derivative. Then for $p\geq 2$ and $x,y\in B$, \begin{equation} \Gamma_{m,n}^{(p)}(x,y)=\int_{B}\dots \int _{B}G_{m,n}(x,z_{1})G_{m,n}(z_{1},z_{2})\dots G_{m,n}(z_{p-1},y)dz_{1} \dots dz_{p-1}, \label{2.1} \end{equation} where $G_{m,n}$ is the Green function of the polyharmonic operator $u\mapsto (-\Delta )^{m}u$, on $B$ with Dirichlet boundary conditions $(\frac{\partial }{\partial \nu })^{j}u=0$, $0\leq j\leq m-1$. Recall that Boggio in \cite{Bo} gave an explicit expression for $G_{m,n}$: For each $x,y$ in $B$, $$ G_{m,n}(x,y)=k_{m,n}{|x-y|}^{2m-n}\int_{1}^{\frac{[ x,y] }{|x-y|}}\frac{(v^{2}-1)^{m-1}}{v^{n-1}}dv, $$ where $k_{m,n}$ is a positive constant. In this section we state some properties of $\Gamma _{m,n}^{(p)}$ and of functions belonging to the Kato class $\mathcal{J}_{m,n}^{(p)}$. These properties are useful for the statements of our existence result, and their proofs can be found in \cite{B}. \begin{proposition} \label{prop2.1} On $B^{2}$, the following estimates hold \begin{equation} \Gamma _{m,n}^{(p)}(x,y)\sim \begin{cases} \frac{(\delta (x)\delta (y))^{m}}{{|x-y|}^{n-2pm}[x,y]^{2m}}, &\text{for }n>2pm, \\[5pt] \frac{(\delta (x)\delta (y))^{m}}{[x,y] ^{2m}} \log(1+\frac{[x,y] ^{2}}{|x-y|^{2}}),&\text{for }n=2pm \\[5pt] \frac{(\delta (x)\delta (y))^{m}}{[x,y] ^{n-2(p-1)m}}, &\text{for }2(p-1)m0$. By \eqref{1.3}, there exists $\alpha >0$ such that for each $x,x'\in B(x_{0},\alpha )\cap B$, we have \begin{align*} &|L\theta (x)-L\theta (x')|\\ & \leq \int_{B}\big|\frac{\Gamma _{m,n}^{(p)}(x,y)}{(\delta (x))^{m}}- \frac{\Gamma _{m,n}^{(p)}(x',y)}{(\delta (x'))^{m}}\big| (\delta (y))^{m}|q(y)|\,dy \\ & \leq \varepsilon +\int_{B\cap B(x_{0},2\alpha )\cap B^{c}(x,2\alpha )}\big|\frac{\Gamma _{m,n}^{(p)}(x,y)}{(\delta (x))^{m}}- \frac{\Gamma _{m,n}^{(p)}(x',y)}{(\delta (x'))^{m}}\big| (\delta (y))^{m}|q(y)|\,dy \\ &\quad +\int_{B\cap B^{c}(x_{0},2\alpha )\cap B^{c}(x,2\alpha )}\big| \frac{\Gamma _{m,n}^{(p)}(x,y)}{(\delta (x))^{m}}-\frac{\Gamma _{m,n}^{(p)}(x',y)}{(\delta (x'))^{m}}\big|(\delta (y))^{m}|q(y)|\,dy \end{align*} Now since for $y\in B^{c}(x,2\alpha )\cap B$, from Proposition \ref{prop2.1}, we have \[ \Gamma _{m,n}^{(p)}(x,y)\preceq (\delta (x)\delta (y))^{m}. \] We deduce that \begin{align*} &\int_{B\cap B(x_{0},2\alpha )\cap B^{c}(x,2\alpha )}|\frac{ \Gamma _{m,n}^{(p)}(x,y)}{(\delta (x))^{m}}-\frac{\Gamma _{m,n}^{(p)}(x',y)}{(\delta (x'))^{m}}|(\delta (y))^{m}|q(y)|\,dy \\ &\preceq \int_{B\cap B(x_{0},2\alpha )}(\delta (y))^{2m}|q(y)|\,dy, \end{align*} which tends by \eqref{2.7} to zero as $\alpha \to 0$. Since for $y\in B^{c}(x_{0},2\alpha )\cap B$, the function $x\mapsto (\frac{\delta (y)}{\delta (x)}) ^{m}\Gamma _{m,n}^{(p)}(x,y)$ is continuous on $B(x_{0},\alpha )\cap B$, by \eqref{2.7} and by the dominated convergence theorem, we have \[ \int_{B\cap B^{c}(x_{0},2\alpha )\cap B^{c}(x,2\alpha )}| \frac{\Gamma _{m,n}^{(p)}(x,y)}{(\delta (x))^{m}}-\frac{\Gamma _{m,n}^{(p)}(x',y)}{(\delta (x'))^{m}}|(\delta (y))^{m}|q(y)|\,dy\to 0 \] as $|x-x'|\to 0$. This proves that the family $L(\mathcal{M}_{q})$ is equicontinuous in $\overline{B}$. It follows by Ascoli's theorem, that $L(\mathcal{M}_{q})$ is relatively compact in $C(\overline{B})$. \end{proof} The next remark will be used to obtain regularity of the solution. \begin{remark} \label{rem3.2}\rm Let $r>n$ and $f$ be a nonnegative measurable function in $L^{r}(B)$. Then $V_{p}f\in C^{2pm-1}(B)$. \end{remark} Indeed, by using the regularity theory of \cite{ADN} (see also \cite[Theorem 5.1]{GS}, and \cite[Theorem IX.32]{Br}), we obtain that $V_{p}f\in W^{2pm,r}(B)$. Furthermore, since $r>n$, then one finds that $V_{p}f\in C^{2pm-1}(B)$ (see \cite[Chap. 7, p.158]{GT}, or \cite[Corollary IX.15]{Br}). \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $K$ be compact in $B$ such that $\gamma :=\int_{K}h(y)dy>0$ and define $r_{0}:=\min_{y\in K}(\delta (y))^{m}>0$. By \eqref{2.3} there exists a constant $c>0$ such that for each $x,y\in B$, \begin{equation} c(\delta (x)\delta (y))^{m}\leq \Gamma _{m,n}^{(p)}(x,y). \label{3.2} \end{equation} By \eqref{1.5} we can find $a>0$ such that $cr_{0}\gamma f(ar_{0})\geq a$. By (H4) and \eqref{2.6}, the function $k\in \mathcal{J }_{m,n}^{(1)}\subset \mathcal{J}_{m,n}^{(p)}$; then it follows from \eqref{2.8} that \[ \delta :=\|V_{p}((\delta (.))^{m}k)\|_{\infty }\leq \|k\|_{m,n,p}<\infty . \] Let $0<\alpha <\frac{1}{\delta }$, then using \eqref{1.6} we can find $\eta >0$ such that for each $t\geq \eta $, $g(t)\leq \alpha t$. Put $\beta :=\sup_{0\leq t\leq \eta } g(t)$. Then we have \begin{equation} 0\leq g(t)\leq \alpha t+\beta ,\text{ for }t\geq 0. \label{3.3} \end{equation} On the other hand, using \eqref{3.2} and \eqref{2.7}, there exists a constant $c_{0}>0$ such that \begin{equation} V_{p}((\delta (.))^{m}k)(x)\geq c_{0}(\delta (x))^{m}. \label{3.4} \end{equation} From (H2), \eqref{2.8} and \eqref{2.6} we derive that \[ \nu :=\|V_{p}(\varphi (.,a(\delta (.))^{m})\| _{\infty }<\infty . \] Put $b=\max \{\frac{a}{c_{0}},\frac{\alpha \nu +\beta }{1-\alpha \delta }\}$ and let $\Lambda $ be the convex set given by \[ \Lambda =\left\{ u\in C(\overline{B}):a(\delta (x))^{m}\leq u(x)\leq V_{p}(\varphi (.,a(\delta (.))^{m})(x)+bV_{p}((\delta (.))^{m}k)(x)\right\} . \] and $T$ be the operator defined on $\Lambda $ by \[ Tu(x)=\int_{B}\Gamma _{m,n}^{(p)}(x,y)[\varphi (y,u(y))+\psi (y,u(y)) ] dy. \] >From \eqref{3.4}, $\Lambda \neq \emptyset $. We will prove that $T$ has a fixed point in $\Lambda $. Indeed, for $u\in \Lambda $, we have by \eqref{1.4}, \eqref{3.2} and the monotonicity of $f$ that \begin{align*} Tu(x) & \geq \int_{B}\Gamma _{m,n}^{(p)}(x,y)\psi (y,u(y))dy \\ & \geq c(\delta (x))^{m}\int_{B}(\delta (y))^{m}h(y)f(u(y))dy \\ & \geq c(\delta (x))^{m}f(ar_{0})r_{0}\int_{K}h(y)dy \\ & \geq a(\delta (x))^{m}. \end{align*} On the other hand, using (H1), \eqref{1.4} and \eqref{3.3}, we deduce that \begin{align*} Tu(x) & \leq V_{p}(\varphi (.,a(\delta (.))^{m}) (x)+\int_{B}\Gamma _{m,n}^{(p)}(x,y)(\delta (y))^{m}k(y)g(u(y))dy \\ & \leq V_{p}(\varphi (.,a(\delta (.))^{m})(x)+\int_{B}\Gamma _{m,n}^{(p)}(x,y)(\delta (y))^{m}k(y)(\alpha u(y)+\beta )dy \\ & \leq V_{p}(\varphi (.,a(\delta (.))^{m})(x)+(\alpha (\nu +b\delta )+\beta )V_{p}((\delta (.))^{m}k) (x) \\ & \leq V_{p}(\varphi (.,a(\delta (.))^{m})(x)+bV_{p}( (\delta (.))^{m}k)(x). \end{align*} Let $v(x)=\varphi (x,a(\delta (x))^{m} / (\delta (x))^{m}$. Then using similar arguments as above, we deduce that for each $u\in \Lambda $ \begin{equation} \begin{gathered} \varphi (.,u)\leq \varphi (.,a(\delta (.))^{m})=(\delta (.))^{m}v , \\ \psi (.,u)\leq g(u)(\delta (.))^{m}k\leq b(\delta (.))^{m}k. \end{gathered} \label{3.5} \end{equation} That is, $\varphi (.,u)+\psi (.,u)\in \mathcal{M}_{(v+bk)(\delta (.))^{m}}.$ Now since by (H2) and (H4), the function $(v+bk)(\delta (.))^{m}\in \mathcal{J}_{m,n}^{(1)}\subset \mathcal{J}_{m,n}^{(p)}$, we deduce from Lemma \ref{lem3.1}, that $T(\Lambda )$ is relatively compact in $C(\overline{B})$. In particular, for all $u\in \Lambda $, $Tu\in C(\overline{B})$ and so $T(\Lambda )\subset \Lambda $. Next, let us prove the continuity of $T$ in $\Lambda $. We consider a sequence $(u_{j})_{j\in \mathbb{N}}$ in $\Lambda $ which converges uniformly to a function $u\in \Lambda $. Then we have \[ |Tu_{j}(x)-Tu(x)|\leq V_{p}[|\varphi (.,u_{j}(.)-\varphi (.,u(.))|+|\psi (.,u_{j}(.))-\psi (.,u(.)|] . \] Now, by \eqref{3.5}, we have \[ |\varphi (.,u_{j}(.)-\varphi (.,u(.))|+|\psi (.,u_{j}(.))-\psi (.,u(.)|\leq 2(1+b)(\delta (.))^{m}(v+k) \] and since $\varphi ,\psi $ are continuous with respect on the second variable, we deduce by \eqref{2.8} and the dominated convergence theorem that \[ \forall x\in B, Tu_{j}(x)\to Tu(x)\quad \text{as }j\to \infty \] Since $T\Lambda $ is relatively compact in $C(\overline{B})$, we have the uniform convergence, namely \[ \|Tu_{j}-Tu\|_{\infty }\to 0\quad \text{as } j\to \infty . \] Thus we have proved that $T$ is a compact mapping from $\Lambda $ to itself. Hence by the Schauder fixed point theorem, there exists $u\in \Lambda $ such that \begin{equation} u(x)=\int_{B}\Gamma _{m,n}^{(p)}(x,y)[\varphi (y,u(y))+\psi (y,u(y)) ] dy. \label{3.6} \end{equation} Using \eqref{3.5}, (H3) and (H5), for each $y\in B$, \begin{equation} \varphi (y,u(y))+\psi (y,u(y))\leq \varphi (y,a(\delta (y) )^{m})+b(\delta (y))^{m}k(y)\in L^{r}(B). \label{3.7} \end{equation} So it is clear that $u$ satisfies (in the sense of distributions) the elliptic differential equation \[ (-\Delta )^{pm} u=\varphi (.,u)+\psi (.,u),\quad\text{in }B. \] Furthermore, by \eqref{3.6}, \eqref{3.7} and Remark \ref{rem3.2}, we deduce that $u\in C^{2pm-1}(B)$. Therefore, using again \eqref{3.6} and \eqref{2.1} we obtain for $j\in \{0,\dots ,p-1\}$, \begin{equation} (-\Delta )^{jm}u(x)=\int_{B}\Gamma _{m,n}^{(p-j)}(x,y)[\varphi (y,u(y))+\psi (y,u(y))] dy. \label{3.8} \end{equation} Using similar arguments as above, we obtain for all $j\in \{0,\dots ,p-1\}$, \begin{equation} a_{j}(\delta (x))^{m}\leq (-\Delta )^{jm}u(x)\leq V_{p-j}(\varphi (.,a_{j}(\delta (.))^{m}))(x)+b_{j}V_{p-j}((\delta (.))^{m}k) (x), \label{3.9} \end{equation} where $a_{j},b_{j}$ are positive constants. Finally, for $j\in \{0,\dots ,p-1\}$, from \eqref{3.9}, \eqref{2.6} and \eqref{2.8}, we have \begin{align*} a_{j}(\delta (x))^{m} & \leq (-\Delta )^{jm}u(x) \\ & \leq (\delta (x))^{m}(\|\frac{\varphi (.,a_{j}(\delta (.))^{m})}{(\delta (.))^{m}}\|_{m,n,p-j}+b_{j}\|k\|_{m,n,p-j})\\ & \preceq (\delta (x))^{m}. \end{align*} So $u$ is the required solution. \end{proof} \begin{example} \label{exp3.3} Let $r>n$, $\lambda 0\quad \text{in }B \\ \lim_{|x|\to 1} \frac{(-\Delta )^{jm}u(x)}{ (1-|x|)^{m-1}}=0,\quad \text{for }0\leq j\leq p-1, \end{gather*} has at least one positive solution, $u\in C^{2pm-1}(B)$, satisfying \[ (-\Delta )^{jm}u(x)\sim (\delta (x))^{m}, \quad \forall j\in \{0,\dots ,p-1\}. \] \end{example} \begin{remark} \label{rem3.4} \rm If $m=1$ and $p\geq 1$, one can obtain similar existence result for \eqref{1.1} on a bounded domain $D\subset \mathbb{R}^{n}$ $(n\geq 2)$ of class $C^{2p,\alpha }$ with $\alpha \in (0,1]$. \end{remark} \subsection*{Acknowledgements} I would like to thank Professor Habib M\^{a}agli for stimulating discussions and useful suggestions. 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