Solutions without any symmetry for semilinear elliptic problems

Abstract

Abstract. We prove the existence of infinitely many solitary waves for the nonlinear Klein-Gordon or Schrödinger equationin R 2 , which have finite energy and whose maximal group of symmetry reduces to the identity.

Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials

“…Therefore, it is to be expected that the spikes cannot be uniformly distributed on a closed curve if a(θ) is not a constant function. A feasible way to answer Question 1 is to develop a theory like [5]. But a more accurate reduction procedure would be required since the mutual angles between the adjacent rays connecting spikes and the origin goes to zero as K tends to infinity.…”

Nonlinear scalar field equations with general nonlinearity

“…Note also the work [28] in which the problem is solved when N = 5 by introducing the O 1 action on H 1 (R N ). Finally in [33] Musso, Pacard and Wei, obtained nonradial solutions for any dimension N ≥ 2, see also [3]. However in all these works stronger assumptions than (f 1) − (f 4) need to be imposed.…”

On the Helmholtz equation and Dancer’s-type entire solutions for nonlinear elliptic equations

“…There is also a deep connection between the solutions of (1) and the constant mean curvature surfaces (CMC). We refer to [2,8,14,15,17] and the references therein for more discussions.…”

“…There are also infinitely many signchanging solutions without any symmetry. See [5,2,7]. Slightly abusing the notation, we use (x, y) to denote the vectors in R m+n , where x represents the first m coordinates.…”