Quasilinear Equations with Neumann Boundary Conditions

TL;DR

This paper establishes the existence of multiple non-constant weak solutions for a class of quasilinear elliptic equations with Neumann boundary conditions, and shows solutions are bounded under certain growth conditions.

Contribution

It provides new multiplicity results for quasilinear equations with Neumann boundary conditions and analyzes solution boundedness under growth assumptions.

Findings

Proves existence of multiple weak solutions.

Demonstrates solutions are bounded under growth conditions.

Extends results to general subcritical and superlinear cases.

Abstract

We prove a multiplicity result for non-constant weak solutions $u \in H^1(\Omega)$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - \lambda u & \text{in } \Omega \\ A(x,u)\nabla u \cdot \eta = 0 & \text{on } \partial \Omega \end{cases} \] where $\lambda \in \mathbb{R}$, $ \Omega$ is a bounded lipschitz domain, $ \eta $ is the outward normal to the boundary $ \partial \Omega $, and $g(x,u)$ is a Carath\'eodory function that satisfies a general subcritical (and superlinear) growth condition. We also prove that any weak solution is bounded under a stronger growth assumption.