September  2003, 2(3): 311-321. doi: 10.3934/cpaa.2003.2.311

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

1. 

Department of Mathematics, Fraţii Buzeşti College, 200352 Craiova, Romania

Received  November 2002 Revised  May 2003 Published  June 2003

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 $\lambda\leq\Lambda$.
Citation: Teodora-Liliana Dinu. Entire solutions of the nonlinear eigenvalue logistic problem with sign-changing potential and absorption. Communications on Pure & Applied Analysis, 2003, 2 (3) : 311-321. doi: 10.3934/cpaa.2003.2.311
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