Existence and Multiplicity for elliptic p-Laplacian problems with critical growth in the gradient

Abstract

We consider the boundary value problem $-Δ_p u = λc(x) |u|^{p-2}u + μ(x) |\grad u|^p + h(x)$, $u\in W_0^{1,p}(\Omega )\cap L^{\infty }(\Omega )$, where $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, is a bounded domain with smooth boundary. We assume $c$, $h\in L^q(\Omega )$ for some $q>\max \{N/p,1\}$ with $c\gneqq 0$ and $\mu \in L^{\infty }(\Omega )$. We prove existence and uniqueness results in the coercive case $\lambda \le 0$ and existence and multiplicity results in the non-coercive case $\lambda >0$. Also, considering stronger assumptions on the coefficients, we clarify the structure of the set of solutions in the non-coercive case.