Uniqueness of solutions for elliptic problems involving the square root of the Laplacian operator

Details Uniqueness of solutions for elliptic problems involving the square root of the Laplacian operator Uniqueness of solutions for elliptic problems involving the square root of the Laplacian operator

2011-09-23

arxiv.org/abs/1109.5146v1

Mathematics

Analysis of PDEs

13 pages. Submitted Sept. 22, 2011

Scientific paper

We examine equations of the form {eqnarray*} \hbox{$(P)_\lambda$}50pt \{{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 where $ \Omega$ is a smooth bounded domain in $ \IR^N$, where $ 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 $ \infty$. We show that the extremal solution $u^*$ associated with the extremal parameter $ \lambda^*$ is unique. We also show that when $f$ is suitably supercritical and when $ \Omega$ is star-shaped with respect to the origin that there is a unique solution for small positive $ \lambda$.

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