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1997 Existence of multiple positive solutions for a semilinear elliptic equation
Yinbing Deng, Yi Li
Adv. Differential Equations 2(3): 361-382 (1997).

Abstract

In this paper, we consider the semilinear elliptic problem $$ -\triangle u+ u=|u|^{p-2}u+ \mu f(x), \quad u \in H^1(\Bbb R^N), \quad N>2. \tag"$(*)_\mu$" $$ For $p> 2$, we show that there exists a positive constant $\mu ^*>0$ such that $(*)_\mu$ possesses a minimal positive solution if $\mu \in (0, \mu ^*)$ and no positive solutions if $\mu > \mu^*$. Furthermore, if $p< \frac{2N}{N-2}$, then $(*)_\mu$ possesses at least two positive solutions for $\mu \in (0, \mu^{*})$, a unique positive solution if $\mu =\mu^*$ and there exists a constant $\mu _{*} >0 $ such that when $ \mu\in (0, \mu_{*})$, problem $(*)_\mu$ possesses at least three solutions. We also obtain some bifurcation results of the solutions at $\mu =0$ and $\mu=\mu^*$.

Citation

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Yinbing Deng. Yi Li. "Existence of multiple positive solutions for a semilinear elliptic equation." Adv. Differential Equations 2 (3) 361 - 382, 1997.

Information

Published: 1997
First available in Project Euclid: 23 April 2013

zbMATH: 1023.35503
MathSciNet: MR1441848

Subjects:
Primary: 35J60
Secondary: 35B05, 35B32

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.2 • No. 3 • 1997
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