Quasilinear Elliptic Cooperative and Competitive Systems

Abstract

We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = λ_1 u_1 + g_{β,1}(u) & \text{in } Ω, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = λ_2 u_2 + g_{β,2}(u) & \text{in } Ω, \\[2mm] u_1 = u_2 = 0 & \text{on } \partialΩ, \end{cases} \] where $\hat{I}{»}_1,\hat{I}{»}_2<\hat{I}{¼}_1$, $\hat{I}{¼}_1$ is the first Dirichlet eigenvalue of the Laplacian, and $\hat{I}\text{c◯}$ is a bounded domain. The nonlinearity is derived from a potential $G_{\hat{I}}^2$ with subcritical growth. We prove the existence of least energy solutions in both the cooperative (${\hat{I}}^2>0$) and competitive (${\hat{I}}^2<0$) regimes. 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.

Keywords: Subcritical nonlinearities, gradient elliptic systems, least energy solutions, mixed cooperation and competition, Dirichlet boundary conditions, quasilinear elliptic equations, nonsmooth critical point theory