Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth

Abstract

We are concerned with Dirichlet problems of the form $${\mathop{\rm div}\nolimits} (|D u|^{p-2}Du)+f(u)=0\ \mbox{ in }Ω,\qquad u=0\ \mbox{ on }\partialΩ, $$ 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 $\gamma (t_1)\ne \gamma (t_2)$ for $t_1\ne t_2$, we prove that, for $\epsilon >0$ small enough, there exists a unique solution of the Dirichlet problem in the domain $\Omega ={\Omega }_{\epsilon }^{\Gamma }=\{(x_1,x_2)\in {\mathbb{R}}^2:\mathrm{dist}((x_1,x_2),\Gamma )<\epsilon \}$, 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 $$\Omega ={\tilde{\Omega }}_{\epsilon ,s}^{\Gamma }=\{(x_1,x_2,y):(x_1,x_2)\in {\Omega }_{\epsilon }^{\Gamma },y\in {\mathbb{R}}^{n-2},|y|<s\}.$$