Non-autonomous quasilinear elliptic equations and Ważewski's principle - Franca, Matteo (Topological Methods in Nonlinear Analysis, 2004)
In this paper we investigate positive radialsolutions of the following equation$$Δ_{p}u+K(r) u|u|^{σ-2}=0$$where $r=|x|$, $x \in {\mathbb R}^n$, $n> p> 1$, $σ=n p/(n-p)$ 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 systemusing a Fowler transform. In [M. Franca,< i> Structure theorems for positive radial solutions of the generalized scalar curvatureequation, when the curvature exhibits a finite number of oscillations< /i> ] the results are obtained using invariant manifold theoryand a dynamical interpretation of the Pohozaev identity;but the restriction $2 n/(n+2) \le p\le 2$ is necessary in order to ensurelocal uniqueness of the trajectories of the system.In this paper we remove this restriction, repeating the proof using a modificationof 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 playedby regular and singular perturbations in this problem, see [M. Franca and R. A. Johnson, < i> Ground states and singular ground states for quasilinearpartial differential equations with critical exponent in the perturbative case< /i> , Adv.Nonlinear Studies].