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Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials

Xianhua Tang, Sitong Chen
2017 Discrete and Continuous Dynamical Systems. Series A  
This paper is dedicated to studying the following Schrödinger-Poisson problem where V (x) is weakly differentiable and f ∈ C(R, R). By introducing some new tricks, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on V and f . Our results generalize and improve the ones in [D. Ruiz, J. Funct. Anal. ] and some related literature. 2010 Mathematics Subject Classification. 35J10, 35J20. Key words and phrases.
more » ... nd phrases. Schrödinger-Poisson problem, ground state solution of Nehari-Pohozaev type, the least energy solutions. 4973 4974 XIANHUA TANG AND SITONG CHEN and in semiconductor theory [5, 25, 26] . For more details in the physical aspects, we refer the readers to [4, 5] . It is well known that Problem (1.1) can be reduced to a nonlinear Schrödinger equation with a nonlocal term. Indeed, as we shall see in Section 2, for any u ∈ H 1 (R 3 ), there exists a unique φ u ∈ D 1,2 (R 3 ) such that − φ = u 2 by using the Lax-Milgram theorem, then inserted into the first equation, gives
doi:10.3934/dcds.2017214 fatcat:spyiu3zulngwfgnh24acflf4cu