Quasilinear Elliptic Cooperative and Competitive Systems

TL;DR

This paper investigates the existence and multiplicity of solutions for a class of quasilinear elliptic systems with potential-driven nonlinearities, employing nonsmooth variational methods due to the non-differentiability of the energy functional.

Contribution

It introduces a novel application of nonsmooth critical point theory to establish solutions for quasilinear elliptic systems with subcritical growth nonlinearities.

Findings

Existence of least energy solutions in cooperative and competitive regimes.

Application of nonsmooth variational methods to non-differentiable functionals.

Solutions obtained for systems with subcritical growth nonlinearities.

Abstract

We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = \lambda_1 u_1 + g_{\beta,1}(u) & \text{in } \Omega, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = \lambda_2 u_2 + g_{\beta,2}(u) & \text{in } \Omega, \\[2mm] u_1 = u_2 = 0 & \text{on } \partial\Omega, \end{cases} \] where $\lambda_1, \lambda_2 < \mu_1$, the first Dirichlet eigenvalue of the Laplacian, and $\Omega$ is a bounded domain. The nonlinearity derives from a potential $G_\beta$ with subcritical growth. Due to the lack of differentiability of the associated energy functional, we employ nonsmooth critical point theory and variational methods based on the concept of weak slope. We prove the existence of least energy solutions in both the cooperative ($\beta > 0$) and competitive ($\beta < 0$) regimes.