Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth

Abstract

We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }\Omega,\qquad u=0\ \mbox{ on }\partial\Omega, $$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$, $n\ge 2$, $1<p<n$ and $f$ is a continuous function with supercritical growth from the viewpoint of the Sobolev embedding. In particular, if $n=2$ and $\gamma:[a,b]\to\mathbb{R}^2$ is a smooth curve such that $\gamma(t_1)\neq\gamma(t_2)$ for $t_1\neq t_2$, we prove that, for $\varepsilon>0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $\Omega=\Omega^\Gamma_\varepsilon=\{(x_1,x_2)\in\mathbb{R}^2\ :\ \mathop{\rm dist}\big((x_1,x_2),\Gamma\big)<\varepsilon\}$, where $\Gamma=\{\gamma(t)\ :\ t\in[a,b]\}$. Moreover, we extend this uniqueness result to the case where $n>2$ and $\Omega$ is, for example, a domain of the type $$ \Omega=\widetilde\Omega^\Gamma_{\varepsilon,s}=\{(x_1,x_2,y)\ :\ (x_1,x_2)\in\Omega^\Gamma_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|<s\}. $$