American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2002, 8 (1) : 267-281. doi: 10.3934/dcds.2002.8.267
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