Infinite multiplicity for inhomogeneous supercritical problem in entire space

TL;DR

This paper proves the existence of infinitely many positive solutions for a supercritical elliptic PDE in the entire space using Lyapunov-Schmidt reduction and asymptotic analysis.

Contribution

It introduces a novel application of Lyapunov-Schmidt reduction to establish infinite multiplicity for supercritical problems in unbounded domains.

Findings

Existence of infinitely many positive solutions in ^n

Solutions tend to zero at infinity

Method applicable to supercritical elliptic equations

Abstract

In this paper, we will prove the existence of infinitely many positive solutions to the following supercritical problem by using the Liapunov-Schmidt reduction method and asymptotic analysis: {ll}\Delta u + u^{p}+f(x)=0, u>0 {in} R^{n}, \lim_{|x|\to\infty}u(x)\to 0.