Topological Methods in Nonlinear Analysis

Non-autonomous quasilinear elliptic equations and Ważewski's principle

Matteo Franca

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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].

Article information

Source
Topol. Methods Nonlinear Anal., Volume 23, Number 2 (2004), 213-238.

Dates
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1464731408

Mathematical Reviews number (MathSciNet)
MR2078191

Zentralblatt MATH identifier
1075.35013

Citation

Franca, Matteo. Non-autonomous quasilinear elliptic equations and Ważewski's principle. Topol. Methods Nonlinear Anal. 23 (2004), no. 2, 213--238. https://projecteuclid.org/euclid.tmna/1464731408


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