The first nontrivial eigenvalue for a system of $p-$Laplacians with Neumann and Dirichlet boundary conditions

Abstract

We deal with the first eigenvalue for a system of two $p-$Laplacians with Dirichlet and Neumann boundary conditions. If \Delta_{p}w=\mbox{div}(|\nabla w|^{p-2}w) stands for the $p-$Laplacian and $\frac{\alpha}{p}+\frac{\beta}{q}=1,$ we consider $$\begin{cases} -\Delta_pu= \lambda \alpha |u|^{\alpha-2} u|v|^{\beta} &\text{ in }\Omega,\\ -\Delta_q v= \lambda \beta |u|^{\alpha}|v|^{\beta-2}v &\text{ in }\Omega,\\ \end{cases}$$ with mixed boundary conditions $$u=0, \qquad |\nabla v|^{q-2}\dfrac{\partial v}{\partial \nu }=0, \qquad \text{on }\partial \Omega.$$ We show that there is a first non trivial eigenvalue that can be characterized by the variational minimization problem $$\lambda_{p,q}^{\alpha,\beta} = \min \left\{\dfrac{\displaystyle\int_{\Omega}\dfrac{|\nabla u|^p}{p}\, dx +\int_{\Omega}\dfrac{|\nabla v|^q}{q}\, dx} {\displaystyle\int_{\Omega} |u|^\alpha|v|^{\beta}\, dx} \colon (u,v)\in \mathcal{A}_{p,q}^{\alpha,\beta}\right\},$$ where $$\mathcal{A}_{p,q}^{\alpha,\beta}=\left\{(u,v)\in W^{1,p}_0(\Omega)\times W^{1,q}(\Omega)\colon uv\not\equiv0\text{ and }\int_{\Omega}|u|^{\alpha}|v|^{\beta-2}v \, dx=0\right\}.$$ We also study the limit of $\lambda_{p,q}^{\alpha,\beta}$ as $p,q\to \infty$ assuming that $\frac{\alpha}{p} \to \Gamma \in (0,1)$, and $\frac{q}{p} \to Q \in (0,\infty)$ as $p,q\to \infty.$ We find that this limit problem interpolates between the pure Dirichlet and Neumann cases for a single equation when we take $Q=1$ and the limits $\Gamma \to 1$ and $\Gamma \to 0$.Comment: 21 pages, 1 figur