Oscillating solutions for nonlinear Helmholtz equations

Abstract

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behaviour at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Kleinâ\u80\u93Gordon or Schrödinger equations with large frequencies

On a fourth‐order nonlinear Helmholtz equation

“…Proof The part (i) is proved exactly as in [, Lemma 4.2], see also [, p.9]. For the part (ii) we argue as in [, Lemma 3.1]. We find a such that so that for sufficiently large is a valid choice by .…”

“…Indeed, one expects that the solutions to will not decay faster than as . This last claim was indeed proved in for all nontrivial radial solutions of a class of nonlinear Helmholtz equations of the form . As a consequence, in this case, the usual energy functional formally associated with is not even well defined on nontrivial solutions.…”

“…We also refer to the previous work of Gutiérrez [22] and the very recent works [28,31] of the third author and his collaborators for several extensions, respectively, to the case where α is replaced by a Z N -periodic potential and to the case of a system. In [30], a sharp decay result for radial solutions was found for the second-order Helmholtz equation.…”

“…As in one may put slightly different assumptions on that still ensure the existence of nontrivial solutions. For instance, replacing the periodicity assumption on by as as in [, Theorem 1.2], the dual variational approach benefits from even better compactness properties that allow to prove the existence of infinitely many solutions via the symmetric mountain pass theorem.…”

“…While the solutions in the former case seem to remain bounded with oscillatory behaviour for all small initial data, the solutions in the latter case seem to be unbounded for most initial data so that we expect ‐solutions as the ones from Theorem only at exceptional initial values. In particular, there is little hope to treat this case by the methods from .…”