Multiplicity and Regularity Results for Quasilinear Elliptic Systems via Nonsmooth Critical Point Theory

TL;DR

This paper proves the existence of infinitely many solutions for a class of quasilinear elliptic systems using nonsmooth critical point theory and establishes bounds for these solutions.

Contribution

It introduces a novel application of nonsmooth critical point theory to quasilinear elliptic systems, demonstrating multiple solutions and bounds.

Findings

Existence of infinitely many weak solutions.

Application of nonsmooth critical point theory.

Establishment of $L^ abla$ bounds for solutions.

Abstract

We study the quasilinear elliptic system \[ -\textbf{div}(A(x,\boldsymbol u)|D\boldsymbol u|^{p-2}D\boldsymbol u) +\frac{1}{p}\nabla_{\boldsymbol s}A(x,\boldsymbol u)|D\boldsymbol u|^p = \boldsymbol g(x,\boldsymbol u) \quad \text{in } \Omega, \qquad \boldsymbol u = 0 \text{ on } \partial\Omega, \] where $p>1$, $\Omega\subset\mathbb R^N$ is a bounded domain with $N>1$, and $\boldsymbol g$ satisfies a subcritical growth condition. In this setting, the associated energy functional is, in general, neither differentiable nor locally Lipschitz in the natural Sobolev space. By exploiting a nonsmooth critical point theory, we prove the existence of infinitely many weak solutions by means of an Equivariant Mountain Pass Theorem. In addition, we establish $L^\infty$-bounds for weak solutions by adapting a Moser-type iteration.