Non-autonomous quasilinear elliptic equations and Ważewski's principle

Matteo Franca

DOI: http://dx.doi.org/10.12775/TMNA.2004.010

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].

Keywords


p-Laplace equations; radial solution; regular/singular ground state; Fowler transform; Ważewski's principle

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