Multiplicity results for ground state solutions of a semilinear equation via abrupt changes in magnitude of the nonlinearity

TL;DR

This paper introduces a class of nonlinear functions with abrupt magnitude changes that guarantee the existence of multiple radially symmetric ground state solutions for a semilinear elliptic equation in higher dimensions.

Contribution

It establishes the existence of at least k ground state solutions using piecewise nonlinearities with controlled abrupt changes in magnitude.

Findings

At least k radially symmetric ground state solutions exist.

Piecewise nonlinear functions with controlled abrupt changes are effective.

The method applies to equations in dimensions greater than 2.

Abstract

Given $k\in\mathbb N$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$\Delta u +f(u)=0,\quad x\in \mathbb R^N, N> 2, $$ has at least $k$ radially symmetric ground state solutions.