Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

TL;DR

This paper proves the uniqueness of extremal solutions for a class of nonlocal elliptic eigenvalue problems, establishing conditions under which solutions are unique for all or small parameters.

Contribution

It demonstrates the uniqueness of extremal solutions in nonlocal elliptic eigenvalue problems and identifies conditions for uniqueness at small parameters.

Findings

Extremal solution $u^*$ is unique for the eigenvalue problem.

Uniqueness holds for small positive $\lambda$ under certain conditions.

Results depend on the supercritical nature of $f$ and geometric properties of $\Omega$.

Abstract

We examine equations of the form {eqnarray*} \{{array}{lcl} \hfill \HA u &=& \lambda g(x) f(u) \qquad \text{in}\ \Omega \hfill u&=& 0 \qquad \qquad \qquad \text{on}\ \pOm, {array}. {eqnarray*} where $ \lambda >0$ is a parameter and $ \Omega$ is a smooth bounded domain in $ \IR^N$, $ N \ge 2$. Here $ g$ is a positive function and $ f$ is an increasing, convex function with $ f(0)=1$ and either $ f$ blows up at 1 or $ f$ is superlinear at infinity. We show that the extremal solution $u^*$ associated with the extremal parameter $ \lambda^*$ is the unique solution. We also show that when $f$ is suitably supercritical and $ \Omega$ satisfies certain geometrical conditions then there is a unique solution for small positive $ \lambda$.