Multiple sign-changing radially symmetric solutions in a general class of quasilinear elliptic equations

Abstract

In this paper we prove that the equation ${-(r^\alpha\phi(|u^{\prime}(r)|)u^{\prime}(r))^{\prime} = \lambda r^\gamma f(u(r)), 0 < r < R}$, 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 ${u^{\prime}(0) =u(R) =0}$. 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 ${\gamma}$.