Existence of decaying entire solutions of a class of semilinear elliptic equations

Kusano, Takaŝi; Noussair, Ezzat S.; Swanson, Charles A.

doi:10.1090/S0002-9939-1988-0929405-1

Existence of decaying entire solutions of a class of semilinear elliptic equations
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by Takaŝi Kusano, Ezzat S. Noussair and Charles A. Swanson PDF

Proc. Amer. Math. Soc. 104 (1988), 1141-1147 Request permission

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.

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