Multiple solutions to the nonlinear Schrödinger equation with a partial confinement

TL;DR

The paper addresses the nonlinear Schrödinger equation with partial confinement, modeled by $-\Delta u+(x_1^2+\cdots+x_m^2)u=f(u)$, and studies standing waves $u$ as critical points of the energy functional $I$ on the Sobolev space $\mathcal{H}$. Using the Nehari manifold and mountain-pass/minimax techniques, it proves the existence of a positive ground state and shows the nonexistence of least-energy sign-changing solutions under partial confinement, complemented by a Hopf-lemma–based moving-planes argument for symmetry. The work further develops a $G$-invariant framework, proving the existence of $G$-invariant ground states with $c_G>0$ and radial symmetry in the $y$-direction, as well as saddle-type nodal solutions governed by odd-symmetry groups. Overall, the results illuminate symmetry, multiplicity, and nodal geometry in NLS with partial confinement and have implications for anisotropic traps in Bose–Einstein condensates.

Abstract

We consider multiple solutions to the nonlinear Schrödinger equation (NLS) with a partial confinement, which is physically relevant to dynamics of the Bose-Einstein condensate. Our study not only verifies the existence of positive ground state solutions and the nonexistence of least energy sign-changing solutions but also sheds light on the symmetry associated with these solutions. A novel finding is the existence of saddle type nodal solutions with their nodal domains intersecting at the origin. Furthermore, we have developed some innovative techniques such as the method of moving planes and the Hopf lemma for nonlinear Schrödinger equations with partial confinement.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1
• Lemma 2.2