N$ and therefore $\varphi^{*}$ is a weak solution of \begin{gather*} - \Delta \varphi^{*} = \lambda_1 (\Omega_0) \varphi^{*} - n(x) (\varphi^{*})^{\rho} \quad \text{in } \Omega\\ \varphi^{*}= 0 \quad \text{on } \partial\Omega . \end{gather*} Moreover, $\varphi^{*}$ is bounded and therefore, by a bootstrap argument $\varphi^{*}$ will be a classical solution of \eqref{eq:elliptic:problem} with $\lambda =\lambda_1 (\Omega_0)$, which contradicts part (i) of Lemma \ref{lem:implicitFThm}, which ends the proof. \end{proof} \begin{remark} It can be shown that $\varphi_{\lambda}$ is globally asymptotically stable for nonnegative nontrivial solutions of \eqref{eq:elliptic:problem}; see \cite{A-P-RB12}. \end{remark} \section{Boundedness and unboundedness of solutions} \label{sec:bound-unbo-solut} The questions are now: What happens as $\lambda \to \lambda_1(\Omega_0)$? Where and how solutions become unbounded? The first that we can say is that the blow-up is a complete blow-up at every point in $\Omega_{0}$. For the for the proofs of the following results, we refer to \cite{A-P-RB12}. \begin{lemma} \label{lem:grow_up_regularset} Assume $K_{0}$ satisfies {\rm (HK)} and let $\{\varphi_\lambda\}$ for $\lambda\in(\lambda_1(\Omega),\lambda_1(\Omega_0))$ denote the family of positive solutions of \eqref{eq:elliptic:problem}. Then \[ \lim_{\lambda \to \lambda_1(\Omega_{0})} \varphi_\lambda(x) = \infty, \quad \text{for all $x\in \Omega_{0}$} . \] \end{lemma} To obtain upper bounds on the solutions outside $\Omega_{0}$ we will use the following Lemma, see \cite{garcia-melian98:_point}. This Lemma analyzes the minimum of a radially symmetric solution of a singular logistic equation with constant coefficients and going to infinity at the boundary, see \cite{keller57:_delta, osserman57:_delta}. \begin{lemma} \label{lem:singular_Dirichlet_pbm} Assume $\rho>1$ and $\lambda, \beta >0$ and consider a ball in $\mathbb{R}^{N}$ of radius $a>0$ and the following singular Dirichlet problem \begin{gather*} - \Delta z = \lambda z -\beta z^{\rho} \quad \text{in } B(0,a) \\ z= \infty \quad \text{on }\partial B(0,a). \end{gather*} Then, there exists a unique positive radial solution, $z_{a}(x)$. Moreover, the solution satisfies $$ \Big( {\lambda \over \beta} \Big)^{1/(\rho -1)} \leq z_{a}(0)= \inf_{B(0,a)} z_{a}(x) \leq \Big( {\lambda (\rho +1) \over 2\beta} + \frac{B}{\beta a^{2}} \Big)^{1/(\rho -1)} $$ for some constant $B=B(\rho,N)>0$, $B$ independent of $\lambda$. \end{lemma} The above Lemma gives a local upper bound. \begin{proposition} \label{prop:boundedness_far_from_K0} Let $x_{0} \in \Omega \setminus K_{0}$ and let $\varphi > 0$ be a stationary solution of \eqref{eq:elliptic:problem} for some $\lambda < \lambda_1(\Omega_0)$. Then there exists $a>0$ and $M>0$ independent of $\lambda$, such that \[ 0\leq \varphi (x) \leq M, \quad \forall x \in B(x_{0}, a). \] \end{proposition} \begin{proof} Let $x_{0} \in \Omega\setminus K_{0}$ and let $a>0$ be such that $B(x_{0}, 3a) \subset \Omega\setminus K_{0}$. Denote $$ \beta =\inf\{ n(x), \; x \in B(x_{0}, 2a)\} >0. $$ For each $y\in B(x_0,a)$, consider $z(x)$ the translation to $B(y, a)$ of the function in Lemma \ref{lem:singular_Dirichlet_pbm}, with $\lambda=\lambda_1(\Omega_0)$. Hence $z(x)$ is a supersolution for $\varphi(x)$ and then \[ \varphi(x) \leq z(x), \quad x\in B(y, a). \] In particular, taking $x=y$, we have $$ \varphi(y)\leq \Big( {\lambda_1(\Omega_0)(\rho +1) \over 2\beta} + \frac{B}{\beta a^{2}}\Big)^{1/(\rho -1)}, \quad \forall y\in B(x_0,a), $$ which proves the result with $M=\left( {\lambda_1(\Omega_0)(\rho +1) \over 2\beta} +\frac{B}{\beta a^{2}} \right)^{1/(\rho -1)}$. \end{proof} Assume now that the two parts $K_1$ and $K_2$ of $K_{0}$ are disjoint. The following result shows that, for $\lambda\to\lambda_1(\Omega_0)$, all solutions of \eqref{eq:elliptic:problem} remain bounded in $K_2$, while they start to grow up in $K_1$. \begin{theorem} \label{thr:progressive_grow_up} Assume $K_{0}$ satisfies {\rm (HK)} and $ K_1 \cap K_2 =\emptyset$. Then the following holds \begin{itemize} \item[(i)] There exists a $\delta>0$ and $M>0$ such that $$ |\varphi_\lambda(x)|\leq M,\quad \forall x:\, d(x,K_2)\leq \delta, \quad \forall \, \lambda\in (\lambda_1(\Omega), \lambda_1(\Omega_0)) . $$ \item[(ii)] For $\lambda \to\lambda_1(\Omega_0)$ all solution of \eqref{eq:elliptic:problem} are bounded on $K_2$. \item[(iii)] If $\lambda \to \lambda_1(\Omega_0)$ then the pointwise limit of the solutions of \eqref{eq:elliptic:problem} is unbounded on $K_1$. \end{itemize} \end{theorem} Now we turn to the case in which $K_1$ and $K_2$ are glued together. First using Lemma \ref{lem:singular_Dirichlet_pbm} we prove the following universal bounds for solutions of \eqref{eq:elliptic:problem}. \begin{lemma} \label{lem:universal_upper_bound} Assume that $n(x)$ satisfies {\rm (Hn)}. Then there exists a constant $A$, independent of $\lambda$ such that for any solution of \eqref{eq:elliptic:problem} we have \[ 0 \leq \varphi (x) \leq h(x) = \Big( \frac{A }{d_0(x)} %\displaystyle\inf_{B\left(x,\frac12 d_0(x)\right)} n} \Big)^{\frac{\gamma +2}{\rho -1}} \] with $d_0(x) = \operatorname{dist}(x,K_{0})$. \end{lemma} The following result will be used further below and gives a criteria to check whether a function that is infinity on a smooth compact set of measure zero, is integrable. As shown below, this criteria depends on the dimension of the set and the rate at which the function diverges on it. \begin{lemma} \label{lem:integral_on_fractal} Assume $K\subset \mathbb{R}^{N}$ is a closed regular $d-$dimensional manifold with $d\leq N-1,$ and consider a function defined on a bounded neighborhood $\Omega$ of $K$ of the form \[ f(x) = \big(dist(x,K)\big)^{-\alpha}\quad \text{for } \alpha >0. \] If $r\geq 1$ satisfies $r\alpha < N-d$, then $f\in L^{r}(\Omega)$. \end{lemma} With all these we can state the following result. \begin{theorem}\label{th:condition:bdd} Assume $K_0$ satisfies {\rm (HK')} and $$ K_1 \cap K_2 \neq \emptyset. $$ Assume $n(x)$ satisfies {\rm (Hn)}. Assume also that \[ \gamma+2<(\rho -1)(N-d) . \] Then, the positive solutions of \eqref{eq:elliptic:problem} remain bounded on compact sets of $\Omega\setminus K_1$. In particular they remain bounded at each point of $K_2\setminus K_1$. \end{theorem} \begin{remark} \rm It is an interesting open problem to determine whether we always obtain that the solution of \eqref{eq:elliptic:problem} are bounded in compact sets of $\Omega\setminus K_1$ or, in the contrary, that we have cases in which $\varphi_\lambda$ becomes infinity in $K_2$ as $\lambda\to \lambda_1(\Omega_0)$. \end{remark} \begin{remark} \rm This work is still in progress, and we refer to \cite{A-P-RB12} for details and more general results, including more general configurations for the set $K_{0}$ and the analysis of the solutions of the parabolic problem associated to \eqref{eq:elliptic:problem}. \end{remark} \subsection*{Acknowledgments} This research was supported by Projects MTM2009-07540, MTM2012-31298 and GR35/10-A, Grupo 920894 BSCH-UCM, Grupo de Investigaci\'on CADEDIF, Spain. \begin{thebibliography}{10} \bibitem{Am76} H. 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