Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems
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Combined effects of logarithmic and superlinear nonlinearities in fractional Laplacian systems Fuliang Wang1 · Hu Die1 · Mingqi Xiang1 Received: 1 May 2020 / Revised: 26 August 2020 / Accepted: 10 November 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we consider the existence and multiplicity of solutions for the following fractional Laplacian system with logarithmic nonlinearity ⎧ (−)s u = λh 1 (x)u ln |u| + ⎪ ⎪ ⎪ ⎨ (−)t v = μh 2 (x)v ln |v| + ⎪ ⎪ ⎪ ⎩ u=v=0
p q p−2 u p+q b(x)|v| |u|
x ∈ ,
q p q−2 v p+q b(x)|u| |v|
x ∈ , x ∈ R N \,
2N where s, t ∈ (0, 1), N > max{2s, 2t}, λ, μ > 0, 2 < p + q < min{ N2N −2s , N −2t }, ⊂ R N is a bounded domain with Lipschitz boundary, h 1 , h 2 , b ∈ C() and (−)s is the fractional Laplacian. When h 1 , h 2 , b are positive functions, the existence of ground state solutions for the problem is obtained. When h 1 , h 2 are sign-changing functions and b is a positive function, two nontrivial and nonnegative solutions are obtained. Our results are new even in the case of a single equation.
Keywords Fractional Laplacian systems · Nehari manifold method · Multiplicity of solutions · Ground state Mathematics Subject Classification 35K55 · 35R11 · 47G20
B
Mingqi Xiang [email protected] Fuliang Wang [email protected] Hu Die [email protected]
1
College of Science, Civil Aviation University of China, Tianjin 300300, People’s Republic of China 0123456789().: V,-vol
9
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F. Wang et al.
1 Introduction and main results This paper deals with the existence and multiplicity of solutions for the following problem involving the fractional Laplacian and logarithmic nonlinearity ⎧ s ⎪ ⎪(−) u = λh 1 (x)u ln |u| + ⎪ ⎨ (−)t v = μh 2 (x)v ln |v| + ⎪ ⎪ ⎪ ⎩ u=v=0
p q p−2 u p+q b(x)|v| |u|
x ∈ ,
q p q−2 v p+q b(x)|u| |v|
x ∈ ,
(1.1)
x ∈ R N \,
where s, t ∈ (0, 1), N > max{2s, 2t}, λ, μ ∈ (0, ∞), 2 < p + q < 2N N min{ N2N −2s , N −2t }, ⊂ R is a bounded domain with Lipschitz boundary, [u]s is the Gagliardo seminorm of u, h 1 , h 2 , b ∈ C() and (−)s is the fractional Laplacian which, up to a normalization constant, is defined for any x ∈ R N as (−) ϕ(x) = 2 lim s
δ→0 R N \Bδ (x)
ϕ(x) − ϕ(y) dy |x − y| N +2s
for any ϕ ∈ C0∞ (R N ). Here Bδ (x) denotes the ball in R N centered at x with radius δ. For basic properties of the fractional Laplacian, we refer the readers to [11] and the references cited there. In recent years, the fractional Laplacian and related nonlocal integro-differential equations have an increasingly applications in many fields, see, for instance, Caffarelli [6], Laskin [17] and Vázquez [29]. The literature on fractional problems is very rich. Here we just refer the readers to [5,12–16,20–23,25,32,33] and the references cited there. To the best of our knowledge, most of the works in the literature studied power type nonlinearity, there are a few papers that deal with the existence and multiplicity of solutions for fractional problems involving logarithmic and power type nonlinearities. In [10], d’Avenia etal
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