Differential and Integral Equations

Nonexistence of signed solutions for a semilinear elliptic problem

Manuel A. del Pino

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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.

Article information

Differential Integral Equations, Volume 8, Number 5 (1995), 1259-1263.

First available in Project Euclid: 20 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B99: None of the above, but in this section


del Pino, Manuel A. Nonexistence of signed solutions for a semilinear elliptic problem. Differential Integral Equations 8 (1995), no. 5, 1259--1263. https://projecteuclid.org/euclid.die/1369056054

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