Quasilinear Equations with Neumann Boundary Conditions

Abstract

We prove a multiplicity result for non-constant weak solutions $u\in H^1(\Omega )$ for the quasilinear elliptic equation \[ \begin{cases} \displaystyle-\text{div}(A(x,u)\nabla u) + \frac{1}{2} D_sA(x,u)\nabla u \cdot \nabla u = g(x,u) - λu & \text{in } Ω\\ A(x,u)\nabla u \cdot η= 0 & \text{on } \partial Ω\end{cases} \] 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 Carathéodory function that satisfies a general subcritical (and superlinear) growth condition. We also prove that any weak solution is bounded under a stronger growth assumption.

arXiv commentKeywords: Subcritical nonlinearities, gradient elliptic systems, Neumann boundary conditions, quasilinear elliptic equations, nonsmooth critical point theory