Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves

Abstract

Weighted inequality theory for fractional integrals is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Basic weighted inequalities are often associated to Hardy, Littlewood and Sobolev [6, 11], Caffarelli, Kohn and Nirenberg [4], respectively to Stein and Weiss [12]. A key attempt in the present paper is to prove a Stein–Weiss inequality with lack of symmetry and variable exponents. We quantify the defect of symmetry of the potential by considering the gap between the minimum and the maximum of the variable exponent. We conclude our work with a section dealing with the existence of stationary waves for a class of nonlocal problems with Choquard nonlinearity and anisotropic Stein–Weiss potential. 2020 Mathematics Subject Classification. 35A23, 47J20, 58E05, 58E35. Funding. The research of Youpei Zhang and Vicenţiu D. Rădulescu was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, project number PCE 137/2021, within PNCDI III. This research is partially supported by the National Natural Science Foundation of China (No. 11971485) and the Fundamental Research Funds for the Central Universities of Central South University (Nos. 2019zzts211, 2021zzts0041). This paper has been completed while Youpei Zhang was visiting University of Craiova (Romania) with the financial support of China Scholarship Council (No. 201906370079). Manuscript received 29th June 2021, accepted 3rd August 2021. ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 960 Youpei Zhang, Xianhua Tang and Vicenţiu D. Rădulescu 1. A weighted anisotropic Stein–Weiss inequality This paper is concerned with the extension and generalization of a classic inequality first considered by Stein and Weiss [12], which is a two-weight counterpart of the Hardy–Littlewood–Sobolev inequality (see [6,11]). In the present work, this new weighted inequality is established in the nonsymmetric and anisotropic setting described by potentials with variable exponent. We offer a new look to anisotropic differential inequalities by controlling the gap between the minimum and the maximum of the variable exponent. This enables us to quantify the defect of symmetry of the potential. Finally, we apply the new anisotropic Stein–Weiss inequality to the study of a nonlocal Choquard problem with variable growth and lack of compactness. The features of this paper are the following: (i) we establish a non-symmetric Stein–Weiss inequality with variable potential; (ii) in order to quantify the defect of symmetry of the potential, we prove more general estimates by considering the gap between the minimum and the maximum of the variable exponent; (iii) the analysis developed in this paper is concerned with the combined effects of a nonhomogeneous differential operator with unbalanced growth and a Choquard nonlinearity with variable exponent; (iv) our analysis combines the nonlocal nature of the Choquard nonlinearity with the local perturbation in the absorption term. Let us first recall the following classical Stein–Weiss inequality [12]. Theorem 1. Let 1 < p, q < +∞, 0 < λ < N , α+β Ê 0 and α+β+λ É N . Then the following properties hold. (i) If 1/p + 1/q + (α+β+λ)/N = 2 and 1− 1/p −λ/N < α/N < 1− 1/p, then there exists a constant C0 =C0(p, q,α,β,λ, N ) <∞ such that ∣∣∣∣∫ RN ∫ RN f (x)g (y) |x|α|x − y |λ|y |β dxdy ∣∣∣∣ÉC0‖ f ‖Lp (RN )‖g‖Lq (RN ), (1) for all f ∈ Lp (RN ), g ∈ Lq (RN ), where C0 is independent of f , g . (ii) For all f ∈ Lp (RN ) there exists a constant C1 = C1(p, q,α,β,λ, N ) < ∞ independent of f , such that ∥∥∥∥∫ RN f (y) |x|α|x − y |λ|y |β dy ∥∥∥∥ Lq (RN ) ÉC1‖ f ‖Lp (RN ), (2) where 1+1/q = 1/p + (α+β+λ)/N and α/N < 1/q < (α+λ)/N . All hypotheses in the previous theorem are sharp. In fact, these conditions are necessary either to ensure integrability or they follow from the scaling of the inequality, which is a special feature of the power-weights case. In the case of radially symmetric functions, the conditionα+βÊ 0 can be relaxed andα+β is allowed to assume negative values, for instanceα+βÊ−(N−1) |p−1−q−1|; see Rubin [10]. In what follows, we set C+(RN ) := {r ∈C (RN ) : 1 < r− := infRN r É r+ := supRN r <+∞} . The main result in this section establishes the following Stein–Weiss inequality with variable exponents. Theorem 2. Let p, q ∈ C+(RN ), f ∈ Lp (RN )∩ Lp (RN ), g ∈ Lq (RN )∩ Lq (RN ), α+β Ê 0 and λ :RN ×RN 7→R be a continuous function such that 0 <λ− := inf RN×RN λÉλ+ := sup RN×RN λ< N and 0 <α+β+λ− Éα+β+λ+ É N . Then the following properties hold. C. R. Mathématique — 2021, 359, n\uf6f0 8, 959-968 Youpei Zhang, Xianhua Tang and Vicenţiu D. Rădulescu 961 (i) There exists a sharp constant C3 = C3(p±, q±,α,β,λ±, N ) <∞, independent of f , g , such that ∣∣∣∣∫ RN ∫ RN f (x)g (y) |x|α|x − y |λ(x,y)|y |β dxdy ∣∣∣∣ÉC3‖ f ‖Lp+(RN )‖g‖Lq+(RN ) +C3‖ f ‖Lp−(RN )‖g‖Lq−(RN ), (3) where max { 1− 1 p+ − λ + N ,1− 1 p− − λ − N } < α N < 1− 1 p− and 1 p(x) + 1 q(y) + α+β+λ(x, y) N = 2, ∀ x, y ∈RN . (ii) Moreover, there exist constants C4 =C4(p+, q+,α,β,λ+, N ) <∞ and C5 =C5(p−, q−,α,β,λ−, N ) <∞, independent of f , such that ∥∥∥∥∫ RN f (y) |x|α|x − y |λ |y |β dy ∥∥∥∥ Lq+(RN ) ÉC4‖ f ‖Lp+(RN ) (4) and ∥∥∥∥∫ RN f (y) |x|α|x − y |λ |y |β dy ∥∥∥∥ Lq−(RN ) ÉC5‖ f ‖Lp−(RN ), (5) where 1+ 1 q+ = 1 p+ + α+β+λ + N , 1+ 1 q− = 1 p− + α+β+λ − N and α N < 1 q+ É 1 q− < α+λ − N . Proof. (i). We first observe that α+β+λ(x, y) É N ( 2− 1 p+ − 1 q+ ) , ∀ x, y ∈RN , =⇒ α+β+λ+ É N ( 2− 1 p+ − 1 q+ ) . On the other hand, we have p+ = sup RN p and q+ = sup RN q, =⇒ there exist {xn}n∈N , { yn } n∈N ⊆RN such that p(xn) → p+, q(yn) → q+ as n →∞. It follows that α+β+λ(xn , yn) → N ( 2− 1 p+ − 1 q+ ) as n →∞. We conclude that 1 p+ + 1 q+ + α+β+λ + N = 2. (6) In a similar way, we obtain 1 p− + 1 q− + α+β+λ − N = 2. (7) Taking into account the elementary inequality 1 |x|α|x − y |λ(x,y)|y |β É 1 |x|α|x − y |λ |y |β + 1 |x|α|x − y |λ |y |β , ∀ x, y ∈R N , C. R. Mathématique — 2021, 359, n\uf6f0 8, 959-968 962 Youpei Zhang, Xianhua Tang and Vicenţiu D. Rădulescu