Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption

Teodora-Liliana Dinu

doi:10.3934/cpaa.2003.2.311

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2003, Volume 2, Issue 3: 311-321. Doi: 10.3934/cpaa.2003.2.311

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Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption

Teodora-Liliana Dinu^1,

1.
Department of Mathematics, Fraţii Buzeşti College, 200352 Craiova

Received: November 2002

Revised: May 2003

Published: September 2003

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Abstract

We are concerned with positive solutions decaying to zero at infinity for the logistic equation $-\Delta u=\lambda ( V(x)u-f(u))$ in $\mathbb R^N$, where $V(x)$ is a variable potential that may change sign, $\lambda$ is a real parameter, and $f$ is an absorbtion term such that the mapping $f(t)/t$ is increasing in $(0,\infty)$. We prove that there exists a bifurcation non-negative number $\Lambda$ such that the above problem has exactly one solution if $\lambda >\Lambda$, but no such a solution exists provided $\lambda\leq\Lambda$.

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Mathematics Subject Classification: 35A05, 35B40, 35J60, 37K50, 92D25.

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