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Systems of coupled Schrödinger equations with sign-changing nonlinearities via classical Nehari manifold approach. (English) Zbl 1416.35235

Summary: We propose existence and multiplicity results for the system of Schrödinger equations with sign-changing nonlinearities in bounded domains or in the whole space \(\mathbb{R}^N\). In the bounded domain we utilize the classical approach via the Nehari manifold, which is (under our assumptions) a differentiable manifold of class \(\mathcal{C}^1\) and the Fountain theorem by Bartsch. In the space \(\mathbb{R}^N\) we additionally need to assume the \(\mathbb{Z}^N\)-periodicity of potentials and our proofs are based on the concentration-compactness lemma by Lions and the Lusternik-Schnirelmann values.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q60 PDEs in connection with optics and electromagnetic theory
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35J47 Second-order elliptic systems
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