Nontrivial solutions of semilinear schrödinger equations on irⁿ and strip-like domains

Abstract

The existence of nontrivial solutions u of the Schrödinger equation

is proved under appropriate assumptions on V and f. It is also considered radially symmetric solutions of (S). Next an existence of nontrivial solutions u of the equation

is considered, where is the first positive eigenvalue of the Laplacian in 0 with Dirichlet boundary conditions. The existence of axially symmetric solutions is also proved. The proofs of results are based on Mountain Pass Theorem and compact embeding lemmata.

Keywords:

• Semilinear Schrödinger equation
• Mountain pass theorem
• radial solutions
• compact embeding lemmata