Differential and Integral Equations

On the number of positive solutions of some semilinear Dirichlet problems in a ball

Adimurthi, Filomena Pacella, and S. L. Yadava

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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.

Article information

Source
Differential Integral Equations, Volume 10, Number 6 (1997), 1157-1170.

Dates
First available in Project Euclid: 1 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367438226

Mathematical Reviews number (MathSciNet)
MR1608057

Zentralblatt MATH identifier
0940.35069

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 34B15: Nonlinear boundary value problems 34C23: Bifurcation [See also 37Gxx] 35B32: Bifurcation [See also 37Gxx, 37K50] 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory

Citation

Adimurthi; Pacella, Filomena; Yadava, S. L. On the number of positive solutions of some semilinear Dirichlet problems in a ball. Differential Integral Equations 10 (1997), no. 6, 1157--1170. https://projecteuclid.org/euclid.die/1367438226


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