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Elliptic Equations and their Synergies

Nehari-type ground state solutions for Schrödinger equations with Hardy potential and critical nonlinearities

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Pages 1315-1335 | Received 14 Dec 2018, Accepted 12 Mar 2019, Published online: 08 Apr 2019
 

ABSTRACT

In this paper, we develop a direct approach to find Nehari-type ground state solutions for the following Schrödinger equation with inverse square potential and critical Sobolev exponent uμ|x|2u+V(x)u=|u|22u+f(x,u),xRN,uH1(RN), where N3, 2=2N/(N2) (the critical Sobolev exponent), 0<μ<(N2)2/4, VC(RN,R) and fC(RN×R,R). Under the conditions N4 and κ+ 214μ/(N2)2>2 and some other mild assumptions on V and f, we prove the existence of a positive ground state solution of Nehari-type to the above problem in the asymptotically periodic case and asymptotically constant case, respectively, with κ(2,2) satisfying inf|x|2r0,t0|t|κ0tf(x,s)ds>0 for some r0>0.

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Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China - NNSF (No: 11601508, No:11571370).

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