On the resonant Lane-Emden problem for the p-Laplacian

Details On the resonant Lane-Emden problem for the p-Laplacian On the resonant Lane-Emden problem for the p-Laplacian

2012-02-17

arxiv.org/abs/1202.3980v1

Mathematics

Analysis of PDEs

Scientific paper

We study the resonant Lane-Emden problem \[\{{array} [c]{rcll}% -\Delta_{p}u & = & \lambda_{p}|u| ^{q-2}u & \text{in}\Omega, u & = & 0 & \text{on}\partial\Omega {array}. \] where $\lambda_{p}$ is the first eigenvalue of the $p$-Laplacian operator $\Delta_{p},$ $p>1,$ and $q$ is close to $p.$ A positive solution $u_{q}$ of this problem is obtained by a suitable scaling of a positive extremal of the Rayleigh quotient associated with the immersion $W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega).$ The main result is the convergence $u_{q}\rightarrow\theta_{p}e_{p}$ in $C^{1}(\bar{\Omega})$ as $q\rightarrow p$, where $\theta_{p}:=\exp(|e_{p}|_{L^{p}(\Omega)}^{-p}% \int_{\Omega}e_{p}|\ln e_{p}|dx)$ and $e_{p}$ is the positive and $L^{\infty}$-normalized first eigenfunction of the $p$-Laplacian. An immediate consequence of this result is that the best constant of the immersion $W_{0}^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ is differentiable at $q=p$ and the value of its derivative is explicitly given in terms of $e_{p}$. The approach allows to handle both sub-linear and super-linear cases simultaneously and it is based on $L^{\infty}$ and $L^{q}$ estimates that are proved in the paper. The main tools used are set level techniques and Picone's Identity. Previous results on the asymptotic behavior (as $q\rightarrow p$) of the positive solutions $u_{\lambda,q}$ of the non-resonant Lane-Emden problem (i.e. with $\lambda_{p}$ replaced by $0<\lambda\neq\lambda_{p}$) are also generalized to the space $C^{1}% (\bar{\Omega}).$

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