Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity

TL;DR

This paper establishes precise exponential decay estimates for solutions of a class of semilinear elliptic equations, revealing new insights into the structure of positive solutions and their critical values, especially under real analytic nonlinearities.

Contribution

It provides sharp decay estimates for solutions of Schrödinger equations with nonlinearities near zero and analyzes the topological structure of positive solutions for real analytic nonlinearities.

Findings

Solutions decay exponentially with precise estimates.

The set of positive solutions is locally path connected.

Existence of discrete critical values for the variational functional.

Abstract

We are concerned with the properties of weak solutions of the stationary Schr\"odinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is H\"older continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schr\"odinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$ we obtain that the set of positive solutions is locally path connected. For a periodic potential $V$ this implies that the standard variational functional has discrete critical values in the low energy range and that a compact isolated set of positive solutions exists, under additional assumptions.