Non-autonomous quasilinear elliptic equations and Ważewski's principle

Abstract

In this paper we investigate positive radial solutions of the following equation ∆pu + K(r)u|u| σ−2 = 0 where r = |x|, x ∈ R n , n > p > 1, σ = np/(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 [9], where structure theorems for positive solutions are obtained for potentials K(r) making a finite number of oscillations. Just as in [9], the starting point is to introduce a dynamical system using a Fowler transform. In [9] the results are obtained using invariant manifold theory and a dynamical interpretation of the Pohozaev identity; but the restriction 2n/(n + 2) ≤ p ≤ 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 < 2n/(n + 2) results similar to the ones obtained in the case 2n/(n + 2) ≤ p ≤ 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].