Multiplicity results for sign changing bound state solutions of a semilinear equation

TL;DR

This paper establishes conditions on the nonlinear function f to guarantee multiple radial bound state solutions with a specified number of zeros for a semilinear elliptic equation in multi-dimensional space.

Contribution

It provides new criteria on f ensuring the existence of multiple solutions with prescribed zero counts, expanding understanding of solution multiplicity in nonlinear PDEs.

Findings

Existence of at least two radial solutions with prescribed zeros.

Conditions on f for multiple solutions with specific properties.

Solutions with initial values in a certain subinterval of (0, ∞).

Abstract

In this paper we give conditions on $f$ so that problem $$ \Delta u +f(u)=0,\quad x\in \mathbb{R}^N, N\ge 2, $$ has at least two radial bound state solutions with any prescribed number of zeros, and such that $u(0)$ belongs to a specific subinterval of $(0,\infty)$. This property will allow us to give conditions on $f$ so that this problem has at least any given number of radial solutions having a prescribed number of zeros.