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Topological Methods in Nonlinear Analysis

Non-autonomous quasilinear elliptic equations and Ważewski's principle
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Non-autonomous quasilinear elliptic equations and Ważewski's principle

Authors

  • Matteo Franca

Keywords

p-Laplace equations, radial solution, regular/singular ground state, Fowler transform, 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.< /p> < p> This paper can be seen as a completion of the work started in [M. Franca, < i> Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations< /i> ], where structure theorems for positive solutions are obtained for potentials $K(r)$ making a finite number of oscillations. Just as in [M. Franca, < i> Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations< /i> ], the starting point is to introduce a dynamical system using a Fowler transform. In [M. Franca, < i> Structure theorems for positive radial solutions of the generalized scalar curvature equation, when the curvature exhibits a finite number of oscillations< /i> ] 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$.< /p> < p> 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, < i> Ground states and singular ground states for quasilinear partial differential equations with critical exponent in the perturbative case< /i> , Adv. Nonlinear Studies].

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Published

2004-06-01

How to Cite

1.
FRANCA, Matteo. Non-autonomous quasilinear elliptic equations and Ważewski’s principle. Topological Methods in Nonlinear Analysis [online]. 1 June 2004, T. 23, nr 2, s. 213–238. [accessed 7.2.2023].
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Issue

Vol 23, No 2 (June 2004)

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