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

Abstract

We consider the boundary value problem \begin{equation} - Δu = λc(x)u+ μ(x) |\nabla u|^2 + h(x), \qquad u \in H^1_0(Ω) \cap L^{\infty}(Ω), \leqno{(P_λ)} \end{equation} where $\Omega \subset {\mathbb{R}}^N,N\ge 3$ 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 $\mu \ge {\mu }_1>0$ for some ${\mu }_1\in \mathbb{R}$. It is known that when $\lambda \le 0$, 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 $]0,+\infty [\times (H_0^1(\Omega )\cap L^{\infty }(\Omega ))$. 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.