Normalized Solutions to Kirchhoff Equation with Nonnegative Potential

TL;DR

This paper investigates the existence of normalized solutions to a Kirchhoff-type equation with nonnegative potential, providing new energy inequalities and bifurcation analysis that extend previous results in the field.

Contribution

The paper introduces a refined analysis for Kirchhoff equations with nonnegative potential, establishing new energy inequalities and bifurcation results for normalized solutions.

Findings

Established existence of bound state solutions under various potential conditions

Derived new energy inequalities for the Kirchhoff problem

Extended previous results to broader classes of potentials

Abstract

This paper is concerned with the existence of solutions to the problem $$-\left(a+ b\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx \right)\Delta u +V(x)u+\lambda u = |u|^{p-2}u,\ \ x \in \mathbb{R}^{N},\ \ \lambda \in \mathbb{R}^{+} $$ where $a, b>0$ are constants, $ V \geq 0$ is a potential, $N \geq 1 $, and $ p \in (2+ \frac{4}{N},2^*$). We use a more subtle analysis to revisit the limited problem($V \equiv 0$), and obtain a new energy inequality and bifurcation results. Based on these observations, we establish the existence of bound state normalized solutions under different assumptions on $V$. These conclusions extend some known results in previous papers.