Infinitely many radial solutions for a p-Laplacian problem with negative weight at the origin

Abstract

We prove the existence of infinitely many sign-changing radial solutions for a Dirichlet problem in a ball defined by the p-Laplacian operator perturbed by a nonlinearity of the form \(W(|x|)g(u),\) where the weight function W changes sign exactly once, \(W(0)<;0\), \(W(1) > 0}, and function g is p-superlinear at infinity. Standard phase plane analysis arguments do not apply here because the solutions to the corresponding initial value problem may blow up in the region where the weight function is negative. Our result extend those in [2] where W is assumed to be positive at 0 and negative at 1.
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Shooting from singularity to singularity and a quasilinear <i>p</i>-Laplace-Beltrami equation with indefinite weight

“…Note that if p = N − 1 we may choose any q 1 , q 2 > p − 1. For future reference we note that due to (6) there exist D > 0, c1 , c2 such that c1 |s| qi+1 ≤ sf (s) ≤ c2 |s| qi+1 , i = 1, 2, for |s| ≥ D.…”

“…For examples of applications to problems with indefinite weight the reader is referred to [12]. For recent results on quasilinear problems with weight see [2,5,6,9,13,15]. For related results on the existence of infinitely many solutions to quasilinear problems see [4,11].…”

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For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of p-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case (p = 2) with positive weight was studied with those in [5] and [6] where radial solutions to a p-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation.

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