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1995 Nonexistence of signed solutions for a semilinear elliptic problem
Manuel A. del Pino
Differential Integral Equations 8(5): 1259-1263 (1995).

Abstract

Let $ \Omega$ be a bounded, smooth domain in ${\Bbb R}^N$, $N\ge 2$. We consider the elliptic boundary value problem $$ \begin{align} \Delta u + u_+^p - u_-^q & =0 \quad\hbox{ in } \ \Omega , \\ u & =0 \quad\hbox{ on } \ \partial \Omega , \end{align} $$ where $1< p < (N+2)/(N-2)$, $0<q<1$, $u_+ = \max\{ u,0\}$, $u_- = -\min\{ u,0\}$. We prove that for certain small domains $\Omega$, no solution to this problem which changes sign in $\Omega$ exists. This answers affirmatively a conjecture raised in [3] where existence of at least one signed solution is established under a largeness condition for the domain.

Citation

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Manuel A. del Pino. "Nonexistence of signed solutions for a semilinear elliptic problem." Differential Integral Equations 8 (5) 1259 - 1263, 1995.

Information

Published: 1995
First available in Project Euclid: 20 May 2013

zbMATH: 0830.35042
MathSciNet: MR1325556

Subjects:
Primary: 35J65
Secondary: 35B99

Rights: Copyright © 1995 Khayyam Publishing, Inc.

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Vol.8 • No. 5 • 1995
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