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The first nontrivial eigenvalue for a system ofp-Laplacians with Neumann and Dirichlet boundary conditions

The first nontrivial eigenvalue for a system ofp-Laplacians with Neumann and Dirichlet boundary conditions

Authors :
Leandro M. Del Pezzo
Julio D. Rossi
Source :
Nonlinear Analysis. 137:381-401
Publication Year :
2016
Publisher :
Elsevier BV, 2016.

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 $\Gamma \to 0$.<br />Comment: 21 pages, 1 figure

Details

ISSN :
0362546X
Volume :
137
Database :
OpenAIRE
Journal :
Nonlinear Analysis
Accession number :
edsair.doi.dedup.....349516c89388d38dd09724ce336e92f1
Full Text :
https://doi.org/10.1016/j.na.2015.09.019