Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schr\"odinger equation

TL;DR

This paper investigates the existence and asymptotic behavior of positive solutions to a class of Schrödinger equations with locally superlinear nonlinearities, extending previous results and considering symmetric potentials.

Contribution

It establishes the existence of positive solutions under new conditions near zero and analyzes their asymptotic behavior as the parameter grows large.

Findings

Existence of positive solutions for trapping and symmetric potentials.

Asymptotic behavior of solutions as ar ty.

Extension of previous existence results.

Abstract

This paper treats the existence of positive solutions of $-\Delta u + V(x) u = \lambda f(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $\lambda > 0$ is a parameter and $f(u)$ satisfies conditions only in a neighborhood of $u=0$. We shall show the existence of positive solutions with potential of trapping type or $\mathcal{G}$-symmetric potential where $\mathcal{G} \subset O(N)$. Our results extend previous results as well as we also study the asymptotic behavior of a family $(u_\lambda)_{\lambda \geq \lambda_0}$ of positive solutions as $\lambda \to \infty$.