Comparison theorems for variational problems and their application to elliptic

I. A. Kuzin

Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 331-346

Résumé

The behavior of PS-sequences for problems of the form $$ \begin{cases} -\sum\limits_{i=1}^N \nabla_ia_i(\mathbf x,u,\nabla u)+b(\mathbf x,u,\nabla u)=0 \quad\text{in}\quad \mathbf R^N, \\ u\to 0 \quad\text{as}\quad |\mathbf x|\to\infty \end{cases} $$ with functions $a_i,b\colon(\mathbf x, \mu,\xi)\mapsto c$ odd with respect to $\mu$, $\xi$ and such that $a_i(\mathbf x, \mu,\xi)\to\bar a_i(\mu,\xi)$, $b(\mathbf x, \mu,\xi)\to\bar b(\mu,\xi)$ as $|\mathbf x|\to\infty$, is studied. On the basis of this, theorems are proved on the existence of $l$ distinct pairs of nontrivial solutions of this problem.