Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth

Abstract

We are concerned with Dirichlet problems of the form $$ {\rm div}(|D u|^{p-2}Du)+f(u)=0\quad \mbox{in }\Omega,\qquad u=0\quad \mbox{on }\partial\Omega, $$ 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 $\gamma\colon [a,b]\to\mathbb{R}^2$ 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 $\Omega=\Omega^\Gamma_\varepsilon=\{(x_1,x_2)\in\mathbb{R}^2 : {\rm dist}\big((x_1,x_2),\Gamma\big)< \varepsilon\}$, 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 $$ \Omega=\widetilde\Omega^\Gamma_{\varepsilon,s}=\{(x_1,x_2,y) : (x_1,x_2)\in\Omega^\Gamma_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|< s\}. $$