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

We study the quasilinear elliptic system $$-\mathbf{\text{div}}(A(x,\mathit{u})\mathrm{\mid }D\mathit{u}{\mathrm{\mid }}^{p-2}D\mathit{u})+\frac{1}{p}{\mathrm{\nabla }}_{\mathit{s}}A(x,\mathit{u})\mathrm{\mid }D\mathit{u}{\mathrm{\mid }}^p=\mathit{g}(x,\mathit{u})\text{in }\mathrm{\Omega },\mathit{u}=0\text{ on }\mathrm{\partial }\mathrm{\Omega },$$ where $p>1$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with $N>1$, and $\mathit{g}$ 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 $L^{\mathrm{\infty }}$-bounds for weak solutions by adapting a Moser-type iteration.