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1997 On the number of positive solutions of some semilinear Dirichlet problems in a ball
Adimurthi, Filomena Pacella, S. L. Yadava
Differential Integral Equations 10(6): 1157-1170 (1997).

Abstract

In this paper we study the problem $-\Delta u = u^{p} + \lambda u^{q}, u > 0$ in $B(1), u = 0$ on $\partial B (1)$ and $\lambda > 0.$ We show that in the case $1 < p \leq {(n+2)/(n-2)}$ there are at most two solutions if $q = 0$, while there are exactly two solutions if $\lambda$ is sufficiently small and $0 < q < 1.$ In the supercritical case $\bigl( p > {(n+2)/(n-2)}\bigr)$ we show the nonexistence of solutions for $1 \leq q < {(n+2)/(n-2)}$ and $\lambda$ sufficiently small. We also consider the problem $- \Delta u = \lambda [ (1 + \alpha u)^{p} + \mu (1 +\alpha u)], u > 0$ in $B(1), u = 0$ on $\partial B(1), \lambda > 0, \alpha > 0$ and $\mu \geq 0.$ We show that for $1 < p \leq {(n+2)/(n-2)}$, there are at most two solutions and, actually, exactly two for $\lambda$ small.

Citation

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Adimurthi. Filomena Pacella. S. L. Yadava. "On the number of positive solutions of some semilinear Dirichlet problems in a ball." Differential Integral Equations 10 (6) 1157 - 1170, 1997.

Information

Published: 1997
First available in Project Euclid: 1 May 2013

zbMATH: 0940.35069
MathSciNet: MR1608057

Subjects:
Primary: 35J65
Secondary: 34B15, 34C23, 35B32, 35P30

Rights: Copyright © 1997 Khayyam Publishing, Inc.

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Vol.10 • No. 6 • 1997
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