Advances in Differential Equations

Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order

F. Bernis, J. García Azorero, and I. Peral

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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.

Article information

Source
Adv. Differential Equations, Volume 1, Number 2 (1996), 219-240.

Dates
First available in Project Euclid: 25 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366896238

Mathematical Reviews number (MathSciNet)
MR1364002

Zentralblatt MATH identifier
0841.35036

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Citation

Bernis, F.; García Azorero, J.; Peral, I. Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differential Equations 1 (1996), no. 2, 219--240. https://projecteuclid.org/euclid.ade/1366896238


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