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

In this paper we prove that the equation $-( r^αϕ(|u'(r)|)u'(r))' = λr^γf(u(r)), ~0<r<R$, where $α, γ, {\bf{R}}$ are given real numbers, $ϕ: (0, \infty) \to (0, \infty)$ is a suitable twice differentiable function, $λ> 0$ 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 $ϕ$ is in the form $ϕ(t) = |t|^β$ for an appropriate constant $γ$.

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