Ground state solutions of Nehari-Pankov type for Schrödinger equations with local super-quadratic conditions

By Xianhua Tang, Sitong Chen, Xiaoyan Lin, Jianshe Yu. Journal of Differential Equations, 2020

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Abstract Consider the semilinear Schrodinger equation { − △ u + V ( x ) u = f ( x , u ) , x ∈ R N , u ∈ H 1 ( R N ) , where both V ( x ) and f ( x , u ) are periodic in x, 0 belongs to a spectral gap of − △ + V , and f ( x , u ) is subcritical and allowed to be super-linear at some x ∈ R N and asymptotically linear at the other x ∈ R N . In the existing works in the literature, it is commonly assumed that lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ uniformly in x ∈ R N , to obtain the existence of ground state solutions or infinitely many geometrically distinct solutions. In this paper, for the first time, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions under the weaker super-quadratic condition lim | u | → ∞ ⁡ ∫ 0 u f ( x , s ) d s u 2 = ∞ , a.e. x ∈ G just for some domain G ⊂ R N .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1d3a59fbb54285c9a4a472af77414f1e1c8fe5f9

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Citations in the corpus

Ground states and geometrically distinct solutions for periodic Choquard-Pekar equations (2020, methodology): “For the Choquard-type equation and related problems, we refer to [11,31,33,44] for the existence of solutions and multiplicity properties, to [13,22,47] for existence of sign-changing solutions, and to [12,32,45,48] for semiclassical solutions.”