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

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].

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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