Existence of Decaying Entire Solutions of a Class of Semilinear Elliptic Equations

Abstract

The main result establishes the existence of a nontrivial nonnegative radial solution $u \in C_{\operatorname {loc} }^2({{\mathbf {R}}^N})$ of a semilinear elliptic eigenvalue problem in ${{\mathbf {R}}^N},N \geq 3$, such that $u(|x|)$ has uniform limit zero as $|x| \to \infty$. Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space $W_0^{1,2}({{\mathbf {R}}^N})$, a considerable departure from standard procedures is required.