Paper detail

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

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 $Ω\subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(Ω)$ for some $p > N$. Also $μ\in L^{\infty}(Ω)$ and $μ\geq μ_1 >0$ for some $μ_1 \in \R$. It is known that when $λ\leq 0$, problem $(P_λ)$ has at most one solution. In this paper we study, under various assumptions, the structure of the set of solutions of $(P_λ)$ assuming that $λ>0$. Our study unveils the rich structure of this problem. We show, in particular, that what happen for $λ=0$ influences the set of solutions in all the half-space $]0,+\infty[\times(H^1_0(Ω) \cap L^{\infty}(Ω))$. 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.