Mathematical Research Letters

Volume 19 (2012)

Number 3

Uniqueness of solutions for a nonlocal elliptic eigenvalue problem

Pages: 613 – 626

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n3.a9

Authors

Craig Cowan (Department of Mathematical Sciences, University of Alabama, Huntsville, Ala., U.S.A.)

Mostafa Fazly (Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada)

Abstract

We examine equations of the form\[\begin{cases}\HA u =\lambda g(x) f(u) & \text{in}\ \Omega \\u=0 & \text{on}\ \pOm,\end{cases}\]where $\lambda>0$ is a parameter and $\Omega$ is a smoothbounded domain in $\IR^N$, $N \ge 2$. Here $g$ is apositive function and $f$ is an increasing, convex functionwith $f(0)=1$ and either $f$ blows up at $1$ or $f$ issuperlinear at infinity. We show that the extremal solution$u^*$ associated with the extremal parameter $\lambda^*$ isthe unique solution. We also show that when $f$ is suitablysupercritical and $\Omega$ satisfies certain geometricalconditions then there is a unique solution for smallpositive $\lambda$.

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