On the multiplicity of weak solutions for a class of coupled quasilinear elliptic systems

Abstract

We study the existence and regularity of weak solutions to the following quasilinear elliptic system: \[ -\mathrm{div}(A_k(x, u_k) |\nabla u_k|^{p_k - 2} \nabla u_k) + \dfrac{1}{p_k} D_s A_k(x, u_k) |\nabla u_k|^{p_k} = g_k(x, u) \quad \text{in } Ω,\quad u_k = 0 \quad \text{on } \partialΩ, \] where $k=1,\dots ,d$, $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with $N\ge 2$, $\mathbf{p}=(p_1,\dots ,p_d)$, $p_k>1$. Using tools from nonsmooth critical point theory, we prove the existence of infinitely many weak solutions in $W_0^{1,\mathbf{p}}(\Omega )\cap L^{\infty }(\Omega ;{\mathbb{R}}^d)$, where $W_0^{1,\mathbf{p}}(\Omega )=W_0^{1,p_1}(\Omega )\times \cdots \times W_0^{1,p_d}(\Omega )$.

Keywords: Subcritical nonlinearities, gradient elliptic systems, Dirichlet boundary conditions, quasilinear elliptic equations, nonsmooth critical point theory