Nodal Solutions for Supercritical Laplace Equations

Abstract

Abstract:In this paper we study radial solutions for the following equationwhere x ∈ R n , n > 2, f is subcritical for r small and u large and supercritical for r 3 large and u small, with respect to the Sobolev critical exponent 2 * = 2n n−2 . The solutions 4 are classified and characterized by their asymptotic behaviour and nodal properties. 5In an appropriate super-linear setting, we give an asymptotic condition sufficient to 6 guarantee the existence of at least one ground state with fast decay with exactly j zeroes 7 for any j ≥ 0. Under the same assumptions, we also find uncountably many ground 8 states with slow decay, singular ground states with fast decay and singular ground states 9 with slow decay, all of them with exactly j zeroes. Our approach, based on Fowler 10 transformation and invariant manifold theory, enables us to deal with a wide family 11 of potentials allowing spatial inhomogeneity and a quite general dependence on u. In 12 particular, for the Matukuma-type potential, we show a kind of structural stability.

On the structure of radial solutions for some quasilinear elliptic equations

“…In particular we recall some of the results contained in [17]. We refer also to [11,19] for the picture in the classical case p = 2. We assume the following hypotheses on the nonlinearity g l .…”

“…The approach for the classical case p = 2 gives a slightly different picture of the phase portrait since, in system (S l ), the term y|y| 2−p p−1 = y is linear when p = 2 (cf. [11,18,19]).…”

“…We focus now our attention on the existence of slow decay solutions following the arguments presented in [11,20].…”

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

“…In fact this analysis is easily extended to any autonomous system (S 0 ) satisfying the following assumption, see [13] for details:…”

“…Proof. We just sketch the proof which is strongly inspired by [13,Section 3] and in particular [13, Lemma 3.14]. The claim concerning u j (τ ),B s j (τ ) follows from construction.…”