Application of Local Linking to Asymptotically Linear Elliptic Equations

Abstract

Existence of a nontrivial solution to a semilinear elliptic equation is established by a variational method. We consider the weak solution to $-\mathrm{\Delta }u=h(x,u)$, where $h(x,u)$ is (asymptotically linear in $u$ both as $u\to 0$ and $u\to \infty$. The proof is based on local linking theory, $(PS{)}^{*}$ condition and approximation by finite dimensional subspaces for the existence of a nontrivial critical point of a $C^1$-class functional.