Advances in Differential Equations

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

Carmen Cortázar, Manuel Elgueta, and Patricio Felmer

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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.

Article information

Source
Adv. Differential Equations Volume 1, Number 2 (1996), 199-218.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366896237

Mathematical Reviews number (MathSciNet)
MR1364001

Zentralblatt MATH identifier
0845.35031

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 34A34: Nonlinear equations and systems, general 34B15: Nonlinear boundary value problems

Citation

Cortázar, Carmen; Elgueta, Manuel; Felmer, Patricio. On a semilinear elliptic problem in $\mathbb R^N$ with a non-Lipschitzian nonlinearity. Adv. Differential Equations 1 (1996), no. 2, 199--218. https://projecteuclid.org/euclid.ade/1366896237.


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