Multiple solvability of certain elliptic problems with critical nonlinearity exponents

I. A. Kuzin

Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 51-56.

Résumé

It is proved that the problem $$ \sum_{i=1}^N\nabla_i(|\nabla u|^{p-2}\nabla_iu)+|u|^{p^*-2}u+\lambda|u|^{q-2}u=0 \text{ in } \Omega, \quad u=0 \text{ on } \partial\Omega, $$ where $\Omega\subset\mathbf{R}^N$ a singly-connected region with an “odd” boundary, $N>p$, and $p^*=Np/(N-p)$ is a critical Sobolev exponent, has, under the appropriate conditions on $\lambda$, $q$ and $N$, no less than $(2N+2)$ nontrivial solutions in $\mathring{W}_{p^1}(\Omega)$.