On a semilinear elliptic problem in $\mathbb R^N$ with a non-Lipschitzian nonlinearity

Abstract

This article is devoted to the study of nonnegative solutions of the semi-linear elliptic equation $$ \Delta u - u^q + u^p = 0 \ \ \text{ in } \; \mathbb{R}^N, $$ where $0 < q < 1 < p < {N+2 \over N-2}$ and $N\geq 3$. We prove the existence of solutions in $H^1(\mathbb{R}^N)$ and that these solutions are compactly supported. Moreover, we show that any solution with the property that $\{x\in \mathbb{R}^N : u(x)>0\}$ is connected, is supported by a ball and is radial. Then we prove that such a solution is unique up to translations. Any other solution is the sum of a finite number of translations of this solution, in such a way that the interior of their supports are mutually disjoint. Existence is proved by a variational argument. The compactness of the support is obtained by comparison. To prove the radial symmetry of solutions we use the Moving Planes device and then, using techniques of ordinary differential equations, we show uniqueness.