Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schr\"{o}dinger equations

TL;DR

This paper proves the existence of multiple solutions for coupled nonlinear bi-harmonic Schrödinger equations using variational methods, including the Mountain Pass Theorem, fibering method, and Lusternik-Schnirel'man theory.

Contribution

It introduces new multiplicity results for coupled bi-harmonic Schrödinger systems via advanced variational techniques.

Findings

Existence of solutions established using Mountain Pass Theorem.

Infinitely many solutions shown through fibering and Lusternik-Schnirel'man theory.

Critical points form a countable family of solutions.

Abstract

We obtain existence and multiplicity results for the solutions of a class of coupled semilinear bi-harmonic Schr\"{o}dinger equations. Actually, using the classical Mountain Pass Theorem and minimization techniques, we prove the existence of critical points of the associated functional constrained on the \emph{Nehari manifold}. Furthermore, we show that using the so-called \emph{fibering method} and the \emph{Lusternik-Schnirel'man theory} there exist infinitely many solutions, actually a countable family of critical points, for such a semiliner bi-hamonic Schr\"{o}dinger system under study in this work.