Abstract
We investigate a class of the nonlinear Schrödinger equation in \( \mathbb {R}^N\)
where \(N\ge 3\), \(\lambda >0\) and \(p\in (2,2^*)\) with \( 2^*=\frac{2 N}{N-2}\). Here, \(V(x)=V_1(x)\) for \(x_1>0\) and \(V(x)=V_2(x)\) for \(x_1<0\), where \(V_1,V_2 \) are periodic in each coordinate direction. By providing a splitting Lemma corresponding to non-periodic external potential, we obtain the existence of ground state solution for the above problem. It is worth to mention that the arguments used in this paper are also valid for the Sobolev subcritical problem studied by Dohnal et al. (Commun Math Phys 308:511–542, 2011).
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The authors would like to extend their sincere gratitude to the referees and the handling editor for their meticulous review of the manuscript and their valuable comments, which have significantly enhanced the quality of the original manuscript.
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This paper is supported by National Natural Science Foundation of China (No. 12371120) and Southwest University graduate research innovation project (No. SWUB23035).
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Jin-Cai Kang: Formal analysis, methodology, writing-original draft. Chun-Lei Tang: Supervision, methodology, formal analysis, writing-review and editing.
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Kang, JC., Tang, CL. Ground States for the Nonlinear Schrödinger Equation with Critical Growth and Potential. Results Math 79, 133 (2024). https://doi.org/10.1007/s00025-024-02166-8
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DOI: https://doi.org/10.1007/s00025-024-02166-8