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

Abstract

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 <p align="center"> $-\Delta u = p(x, u) + f(\theta, x, u)\quad $ on $\Omega$ <p align="left" class="times"> <p align="center"> $u = 0\quad$ on $\partial \Omega$ <p align="left" class="times"> where $p(x, \cdot)$ is odd and $f$ is a perturbative term. An application of this result is the problem <p align="center"> $-\Delta u = \lambda |u|^{q-1}u + |u|^{p-1}u + f\quad$ on $\Omega$ <p align="left" class="times"> <p align="center"> $u = u_0\quad$ on $\partial \Omega$ <p align="left" class="times"> 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.