Non-autonomous quasilinear elliptic equations and Ważewski's principle

Abstract

In this paper we investigate positive radial solutions of the following equation $$ \Delta_{p}u+K(r) u|u|^{\sigma-2}=0 $$ where $r=|x|$, $x \in {\mathbb R}^n$, $n> p> 1$, $\sigma =n p/(n-p)$ is the Sobolev critical exponent and $K(r)$ is a function strictly positive and bounded.

This paper can be seen as a completion of the work started in [M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations], where structure theorems for positive solutions are obtained for potentials $K(r)$ making a finite number of oscillations. Just as in [M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations], the starting point is to introduce a dynamical system using a Fowler transform. In [M. Franca, Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations] the results are obtained using invariant manifold theory and a dynamical interpretation of the Pohozaev identity; but the restriction $2 n/(n+2) \le p\le 2$ is necessary in order to ensure local uniqueness of the trajectories of the system. In this paper we remove this restriction, repeating the proof using a modification of Ważewski's principle; we prove for the cases $p> 2$ and $1< p< 2 n/(n+2)$ results similar to the ones obtained in the case $ 2 n/(n+2) \le p\le 2$.

We also introduce a method to prove the existence of Ground States with fast decay for potentials $K(r)$ which oscillates indefinitely. This new tool also shed some light on the role played by regular and singular perturbations in this problem, see [M. Franca and R. A. Johnson, Ground states and singular ground states for quasilinear partial differential equations with critical exponent in the perturbative case, Adv. Nonlinear Studies].