Bounded Positive Solutions of Semilinear Schrödinger Equations

Abstract

The Schrödinger equation (1) $\Delta u + f(x,u) = 0$ is considered in an exterior domain $\Omega $ in $R^n ,n \geqq 2$, where f is Hölder continuous and nonnegative and ${f(x,u)} / {u}$ is majorized above and below by nonnegative functions $g(| x |,u)$ which are monotone in u for $u > 0,| x | \geqq 0$. Conditions on f are found which are necessary and sufficient for (1) to have a uniformly positive bounded solution in $\Omega \subset R^2 $, and corresponding results in $\Omega \subset R^2 $, $n \geqq 3$. Such theorems constitute the only characterizations discovered to date of partial differential equations possessing positive solutions with specified behavior at $\infty $.