January  2002, 8(1): 267-281. doi: 10.3934/dcds.2002.8.267

Deformation from symmetry and multiplicity of solutions in non-homogeneous problems

1. 

Department of Mathematics, The University of British Columbia, Vancouver, B.C. V6M 3X7, Canada

2. 

Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2, Canada

Received  February 2001 Revised  September 2001 Published  October 2001

A general theorem on the multiplicity of critical points for non-invariant deformations of symmetric functionals is established, using a method introduced by Bolle [5]. This result is used to find conditions sufficient for the existence of multiple solutions of semi-linear elliptic partial differential equations of the form

$-\Delta u = p(x, u) + f(\theta, x, u)\quad $ on $\Omega$

$u = 0\quad$ on $\partial \Omega$

where $p(x, \cdot)$ is odd and $f$ is a perturbative term. An application of this result is the problem

$-\Delta u = \lambda |u|^{q-1}u + |u|^{p-1}u + f\quad$ on $\Omega$

$u = u_0\quad$ on $\partial \Omega$

where $\Omega$ is a smooth, bounded, open subset of $\mathbf R^n (n \geq 3), \lambda > 0, 1\leq q < p, f \in C(\bar \Omega, \mathbf R)$ and $u_0\in C^2(\partial \Omega, \mathbf R)$. It is proven that this equation has an infinite number of solutions for $p < \frac{n+1}{n-1}$ and that for any sub-critical $p$ i.e., $p < \frac{n+2}{n-2}$, there are as many solutions as we like, provided $||f||_{frac{p+1}{p}}$ and $||u_0||_{p+1}$ are small enough.

Citation: Christine Chambers, Nassif Ghoussoub. Deformation from symmetry and multiplicity of solutions in non-homogeneous problems. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 267-281. doi: 10.3934/dcds.2002.8.267
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