Solutions of Schrödinger equations with symmetry in orientation preserving tetrahedral group

Abstract

We consider the nonlinear Schrödinger equationΔu=(1+εV1(|y|))u−|u|p−1uinRN,N≥3,p∈(1,N+2N−2). The phenomenon of pattern formation has been a central theme in the study of nonlinear Schrödinger equations. However, the following nonexistence of O(N) symmetry breaking solution is well-known: if the potential function is radial and nondecreasing, any positive solution must be radial. Therefore, solutions of interesting patterns can only exist after violating the assumptions.O(N) symmetry breaking solutions have been presented by Wei and Yan [20]. Symmetry groups of regular polygons describe their solution patterns. Ever since work of Wei and Yan, there have been substantial generalizations but solutions with higher dimensional symmetry have not been constructed. In this study, the existence of nonradial solutions whose symmetry group is a discrete subgroup of O(3), more precisely, the orientation-preserving regular tetrahedral group is shown.