Multiple Sign Changing Radially Symmetric Solutions in a General Class of Quasilinear Elliptic Equations

TL;DR

This paper proves the existence of infinitely many sign-changing solutions for a broad class of quasilinear elliptic equations with radial symmetry, extending previous results to more general nonlinearities.

Contribution

It introduces fixed point and shooting methods to establish multiple solutions for a general class of quasilinear elliptic equations, broadening earlier specific cases.

Findings

Existence of infinitely many sign-changing solutions.

Solutions satisfy boundary conditions u'(0)=u(R)=0.

Extension of previous results to more general nonlinear functions.

Abstract

In this paper we prove that the equation $ -( r^\alpha\phi(|u'(r)|)u'(r))' = \lambda r^\gamma f(u(r)), ~0<r<R$, where $\alpha, \gamma, {\bf{R}}$ are given real numbers, $\phi : (0, \infty) \to (0, \infty)$ is a suitable twice differentiable function, $\lambda > 0$ is a real parameter and $f:{\bf{R}}\to{\bf{R}}$ is continuous, admits an infinite sequence of sign-changing solutions satisfying $u'(0) =u(R) =0$. The function $f$ is required to satisfy $tf(t)>0$ for $ t\neq 0$. Our technique explores fixed point arguments applied to suitable integral equations and shooting arguments. Our main result extends earlier ones in the case $\phi$ is in the form $\phi(t) = |t|^{\beta}$ for an appropriate constant $\gamma$.