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1996 Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order
F. Bernis, J. García Azorero, I. Peral
Adv. Differential Equations 1(2): 219-240 (1996).

Abstract

In this paper we consider the equation $\Delta^2 u = \lambda |u|^{q-2} u + |u|^{2^*-2} u\equiv f(u)$ in a smooth bounded domain $\Omega\subset\mathbf{R}{N}$ with boundary conditions either $u|_{\partial\Omega} =\frac{\partial u}{\partial n}|_{\partial \Omega}=0$ or $u|_{\partial\Omega}=\Delta u|_{\partial \Omega}=0$, where $N>4$, $ 1< q <2, \,\lambda>0$ and $2^* = 2N/(N-4)$ We prove the existence of $\lambda_0$ such that for $0<\lambda<\lambda_0$ the above problems have infinitely many solutions. For the problem with the second boundary conditions, we prove the existence of a positive solution also in the supercritical case, i.e., when we have an exponent larger than $2^*$. Moreover, in the critical case, we show the existence of at least two positive solutions.

Citation

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F. Bernis. J. García Azorero. I. Peral. "Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order." Adv. Differential Equations 1 (2) 219 - 240, 1996.

Information

Published: 1996
First available in Project Euclid: 25 April 2013

zbMATH: 0841.35036
MathSciNet: MR1364002

Subjects:
Primary: 35J65
Secondary: 35B05 , 58E05

Rights: Copyright © 1996 Khayyam Publishing, Inc.

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Vol.1 • No. 2 • 1996
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