Multiple solvability of certain elliptic problems with critical nonlinearity exponents

I. A. Kuzin

Branch for Theoretical Problems, Russian Academy of Sciences

Abstract: 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 $\mathring{W}_{p^1}(\Omega)$.

UDC: 517

Received: 13.11.1991

 English version: Mathematical Notes, 1992, 52:1, 668–672

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