Existence of Solutions for Fractional-Choquard Equation with a Critical Exponential Growth in $${\mathbb {R}}^N$$ R
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Existence of Solutions for Fractional-Choquard Equation with a Critical Exponential Growth in RN Caisheng Chen Abstract. In this work, we study the existence of solutions to the fractional-Choquard equation (−Δ)sν u + V (x)|u|ν−2 u = (Iα ∗ |u|q )|u|q−2 u + h(u),
x ∈ RN ,
(0.1)
where ν = Ns , 0 < s < 1, N ≥ 2, V (x) is a positive and bounded function in RN , Iα is the Riesz potential, qs > N and the continuous N function h(u) behaves like exp(α0 |u| N −s ) growth. Using the symmetric rearrangement method with some special techniques and symmetric mountain pass lemma, we prove the existence of infinitely many solutions for (0.1) in W s,ν (RN ). Mathematics Subject Classification. 35Q55, 35R09, 35J91. Keywords. Fractional-Choquard equation, symmetric rearrangement method, critical exponential growth.
1. Introduction and Main Result Recently, Pucci et al. in [33] studied the existence of solutions for the fractional p-Laplacian–Choquard–Kirchhoff equations: )[(−Δ)sp u + V (x)|u|p−2 u] (a + bup(θ−1) s ∗
∗
= λf (x, u) + (κμ (x) ∗ |u|pμ,s )|u|pμ,s −2 u, x ∈ RN , [1, p∗s /p), p∗s
pN N −ps , N
ps, p∗μ,s
(1.1)
p∗s (1
where θ ∈ = > = − μ/2N ), κμ (x) = −μ |x| , μ ∈ (0, N ). The function f (x, u) is nonlinear in u and the power of u is less than p∗s , and the potential function V (x) satisfies
(V1 ) V (x) : RN → R+ is continuous and there exists V0 > 0 such that inf x∈RN V (x) ≥ V0 ; (V2 ) there exists h > 0 such that lim|y|→∞ meas{x ∈ Bh (y) : V (x) ≤ c} = 0 for all c > 0. 0123456789().: V,-vol
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C. Chen
MJOM
The condition (V2 ) was first proposed in [5] to overcome the lack of compactness in Sobolev embedding W 1,p (RN ) → Lq (RN ). More recently, Xiang et al. in [40] studied the following fractional Schr¨ odinger–Kirchhoff type equation with the Trudinger–Moser nonlinearity f (x, u): M (uνW s,ν (RN ) )[(−Δ)sν u + V (x)|u|ν−2 u] = f (x, u) + λh(x)|u|p−2 u, x ∈ RN ,
(1.2) N s
s,ν
N
where ν = is the borderline case of the Sobolev embedding W (R ) → q N L (R ). A typical case of M is given by M (t) = a + btθ−1 , t ≥ 0 and V (x) satisfies the conditions (V1 ) − (V2 ). Obviously, when θ = 1, p = 2 and V (x) ≡ ω > 0, λ = 0, Eq. (1.1) is reduced to the fractional-Choquard equation (−Δ)s u + ωu = (Iα ∗ |u|q )|u|q−2 u,
x ∈ RN ,
(1.3)
which was studied by d’Aenia, Siciliano and Squassina in [16]. They obtained the regularity, existence, nonexistence, symmetry and decay properties of the corresponding solutions for (1.3). For s = 1/2, problem (1.3) has been used to model the dynamics of pseudo-relativistic boson stars. Indeed, in [18], the following equation √ −Δ u + u = (I2 ∗ |u|2 )u, u ∈ H 1/2 (R3 ), u > 0, (1.4) was studied. In [14], the existence and concentration of a single-spike solution for the generalized pseudo-relativistic Hartree equation −ε2 Δ + m u + V u = (Iα ∗ |u|q )|u|q−2 u, x ∈ RN (1.5) was obtained. We refer the interested readers to [14,15,22,39] and the references therein for more results on the pseudo-relativistic Hartree equation. The Choquard equation involving a fractio
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