Uniqueness of solutions for nonlinear Dirichlet problems with supercritical growth

TL;DR

This paper proves the uniqueness of solutions for certain nonlinear Dirichlet problems with supercritical growth, especially in thin tubular domains around curves, extending results from two dimensions to higher dimensions.

Contribution

It establishes the uniqueness of solutions for supercritical nonlinear Dirichlet problems in thin domains around curves, extending previous results to higher dimensions.

Findings

Unique solutions exist in small tubular neighborhoods of curves.

Extension of uniqueness results from 2D to higher dimensions.

Solutions are unique for supercritical growth nonlinearities.

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 $$ \Omega=\widetilde\Omega^\Gamma_{\varepsilon,s}=\{(x_1,x_2,y)\ :\ (x_1,x_2)\in\Omega^\Gamma_\varepsilon, \ y\in\mathbb{R}^{n-2},\ |y|<s\}. $$