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

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.