Ground state solution for fractional problem with critical combined nonlinearities

Xu, Er-Wei and Sun, Hong-Rui (2023) Ground state solution for fractional problem with critical combined nonlinearities. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (38). pp. 1-18. ISSN 1417-3875

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Abstract

This paper is concerned with the following nonlocal problem with combined critical nonlinearities (−∆)su = −α|u|q−2u+ βu+γ|u|2∗ s−2u in Ω, u =0 inRN\Ω, where s ∈ (0,1), N > 2s, Ω ⊂ RN isabounded C1,1 domain with Lipschitz boundary, α is a positive parameter, q ∈ (1,2), β and γ are positive constants, and 2∗ s = 2N/(N−2s) is the fractional critical exponent. For γ > 0, if N ⩾ 4s and 0 < β < λ1,s, or N > 2s and β ⩾ λ1,s, we show that the problem possesses a ground state solution when α is sufficiently small.

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