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 ∈ C loc 2 ( R N ) u \in C_{\operatorname {loc} }^2({{\mathbf {R}}^N}) of a semilinear elliptic eigenvalue problem in R N , N ≥ 3 {{\mathbf {R}}^N},N \geq 3 , such that u ( | x | ) u(|x|) has uniform limit zero as | x | → ∞ |x| \to \infty . Asymptotic decay estimates and necessary conditions are obtained. Since such solutions do not exist in the space W 0 1 , 2 ( R N ) W_0^{1,2}({{\mathbf {R}}^N}) , a considerable departure from standard procedures is required.