Multiplicity results in the non-coercive case for an elliptic problem with critical growth in the gradient

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Abstract

We consider the boundary value problem \begin{equation} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \qquad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega), \leqno{(P_{\lambda})} \end{equation} where $\Omega \subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(\Omega)$ for some $p > N$. Also $\mu \in L^{\infty}(\Omega)$ and $\mu \geq \mu_1 >0$ for some $\mu_1 \in \R$. It is known that when $\lambda \leq 0$, problem $(P_{\lambda})$ has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of $(P_{\lambda})$ assuming that $\lambda>0$. Our study unveils the rich structure of this problem. We show, in particular, that what happen for $\lambda=0$ influences the set of solutions in all the half-space $]0,+\infty[\times(H^1_0(\Omega) \cap L^{\infty}(\Omega))$. Most of our results are valid without assuming that $h$ has a sign. If we require $h$ to have a sign, we observe that the set of solutions differs completely for $h\gneqq 0$ and $h\lneqq 0$. We also show when $h$ has a sign that solutions not having this sign may exists. Some uniqueness results of signed solutions are also derived. The paper ends with a list of open problems.