Differential and Integral Equations

Nonexistence and uniqueness for biharmonic problems with supercritical growth and domain geometry

Saïma Khenissy

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Abstract

We prove nonexistence and uniqueness results of positive solutions for biharmonic supercritical equations $\Delta ^2 u = f(u)$ under Navier boundary conditions on a smooth bounded domain $\Omega \subset \mathbb{R}^N$. The results stand for suitable supercritical nonlinearities $f$ with some geometrical conditions on $ \Omega.$ We define the $h$-starlikeness of $\Omega$ and a classifying number $M(\Omega)$. This allows us to define a generalized critical exponent for these domains which play the role of the classical exponent $\frac{N+4}{N-4}$. Our approach is based on Rellich-Pohozaev type estimates. In particular, we construct some $h $-starlike domains which are topologically nontrivial where our results can apply.

Article information

Source
Differential Integral Equations, Volume 24, Number 11/12 (2011), 1093-1106.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356012878

Mathematical Reviews number (MathSciNet)
MR2866013

Zentralblatt MATH identifier
1249.35089

Subjects
Primary: 35J40: Boundary value problems for higher-order elliptic equations 35J30: Higher-order elliptic equations [See also 31A30, 31B30] 35J60: Nonlinear elliptic equations

Citation

Khenissy, Saïma. Nonexistence and uniqueness for biharmonic problems with supercritical growth and domain geometry. Differential Integral Equations 24 (2011), no. 11/12, 1093--1106. https://projecteuclid.org/euclid.die/1356012878


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