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

TL;DR

This paper investigates the existence, uniqueness, and multiplicity of solutions for a class of elliptic p-Laplacian boundary value problems with critical growth in the gradient, depending on the parameter λ and coefficient conditions.

Contribution

It provides new existence and multiplicity results for elliptic p-Laplacian problems with critical gradient growth, including a detailed analysis of solution structure under various coefficient assumptions.

Findings

Existence and uniqueness for λ ≤ 0

Existence and multiplicity for λ > 0

Detailed solution structure under stronger assumptions

Abstract

We consider the boundary value problem $-\Delta_p u = \lambda c(x) |u|^{p-2}u + \mu(x) |\grad u|^p + h(x)$, $u \in W^{1,p}_0(\Omega) \cap L^{\infty}(\Omega)$, where $\Omega \subset \mathbb R^N$, $N \geq 2$, 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 \leq 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.