Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients

TL;DR

This paper investigates the existence and multiplicity of solutions for a nonlinear elliptic boundary value problem with critical growth in the gradient and sign-changing coefficients, using variational and topological methods.

Contribution

It provides necessary and sufficient conditions for solution existence when , and establishes existence and multiplicity results for , employing a novel change of variables and combined analytical techniques.

Findings

Unique solution for  under specific conditions.

Existence of multiple solutions for  when  solutions exist.

Conditions linking coefficient signs to solution multiplicity.

Abstract

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_{\lambda}$} -\Delta u = c_{\lambda}(x) u + \mu |\nabla u|^2 + h(x)\,, \quad u \in H_0^1(\Omega) \cap L^{\infty}(\Omega)\,, \end{equation} where $c_{\lambda}$ and $h$ belong to $L^q(\Omega)$ for some $q > N/2$, $\mu$ belongs to $\mathbb{R} \setminus \{0\}$ and we write $c_{\lambda}$ under the form $c_{\lambda}:= \lambda c_{+} - c_{-}$ with $c_{+} \gneqq 0$, $c_{-} \geq 0$, $c_{+} c_{-} \equiv 0$ and $\lambda \in \mathbb{R}$. Here $c_{\lambda}$ and $h$ are both allowed to change sign. As a first main result we give a necessary and sufficient condition which guarantees the existence of a unique solution to \eqref{Plambda-Abstract-ch3} when $\lambda \leq 0$. Then, assuming that $(P_0)$ has a solution, we prove existence and multiplicity results for $\lambda > 0$. Our proofs rely on a suitable change of variable of type $v = F(u)$ and the combination of variational methods with lower and upper solution techniques.