In this paper we investigate positive radial solutions of the following equation: egin{equation*} Delta_{p}u+K(r) u|u|^{sigma-2}=0 end{equation*} where $r=|x|$, $x in RR^n$, $n>p>1$, $sigma =rac{n p}{n-p}$ 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 cite{F5}, where structure theorems for positive solutions are obtained for potentials $K(r)$ making a finite number of oscillations. Just as in cite{F5}, the starting point is to introduce a dynamical system using a Fowler transform. In cite{F5} the results are obtained using invariant manifold theory and a dynamical interpretation of the Pohozaev identity; but the restriction $ rac{2 n}{n+2} le ple 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 Wazewski's principle; we prove for the cases $p>2$ and $1< rac{2 n}{n+2}$ results similar to the ones obtained in the case $ rac{2 n}{n+2} le ple 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 [10]
Non-Autonomous Quasilinear Elliptic Equations and Wazewski's principle
M. FRANCA
2004
Abstract
In this paper we investigate positive radial solutions of the following equation: egin{equation*} Delta_{p}u+K(r) u|u|^{sigma-2}=0 end{equation*} where $r=|x|$, $x in RR^n$, $n>p>1$, $sigma =rac{n p}{n-p}$ 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 cite{F5}, where structure theorems for positive solutions are obtained for potentials $K(r)$ making a finite number of oscillations. Just as in cite{F5}, the starting point is to introduce a dynamical system using a Fowler transform. In cite{F5} the results are obtained using invariant manifold theory and a dynamical interpretation of the Pohozaev identity; but the restriction $ rac{2 n}{n+2} le ple 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 Wazewski's principle; we prove for the cases $p>2$ and $1< rac{2 n}{n+2}$ results similar to the ones obtained in the case $ rac{2 n}{n+2} le ple 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 [10]I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
