Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

2011/09/23 by Cowan, Craig, Fazly, Mostafa
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1109.5146

Abstract

We examine equations of the form eqnarray* \arraylcl \hfill \HA u &=& λg(x) f(u) in Ω\hfill u&=& 0 on \pOm, array. eqnarray* where λ>0 is a parameter and Ω is a smooth bounded domain in \IR^N, N ≥ 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 λ^* is the unique solution. We also show that when f is suitably supercritical and Ω satisfies certain geometrical conditions then there is a unique solution for small positive λ.

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