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In this paper, we consider a singular Adams' inequality with logarithmic weights in the unit ball of $ \mathbb {R}^4 $ R 4 . Our results extend the results of Zhu and Wang [Adams' inequality with logarithmic weights in $ \mathbb {R}^4 $ R 4 . Proc Amer Math Soc. 2021;149(8):3463–3472] on Adams' inequality with logarithmic weights to singular case. Then, we study the existence of solutions for some weighted mean field equations, relying on variational methods and the singular Adams' inequality with logarithmic weights we previously established. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution p, q-Laplacian equations u t = d i v (| ∇ u | p − 2 ∇ u) + u α ∫ Ω v m d x and v t = d i v (| ∇ v | q − 2 ∇ v) + v β ∫ Ω u n d x with 1 (p − 1 − α) (q − 1 − β) , there exist suitable initial data such that vanishing solutions exist. If m n < (p − 1 − α) (q − 1 − β) , we find the explicit scopes of initial data such that the solutions could not vanish, which complete the corresponding classifications of solutions in Math. Methods Appl. Sci. 39 (2016) 1325–1335 and Appl. Math. Comp. 259 (2015) 587–595, respectively. For the critical case m n = (p − 1 − α) (q − 1 − β) , the solutions vanish in finite time with small initial data. [ABSTRACT FROM AUTHOR]

In this paper, as an attempt we consider the linear self-interacting diffusion driven by an α-stable motion, which is the solution to the equation X t α = M t α − θ ∫ 0 t ∫ 0 s (X s α − X r α) d r d s + ν t , where θ ≠ 0 , ν ∈ R and M α is an α-stable motion on R ( 0 < α ≤ 2). The process is an analogue of the self-attracting diffusion (see Durrett-Rogers, Prob. Theory Related Fields92 (1992), 337–349, and Cranston-Le Jan, Math. Ann.303 (1995), 87–93.). The main object of this paper is to prove some limit theorems associated with the solution process X α for 1 2 < α ≤ 2. When θ > 0 we show that ψ α (t) (X t α − X ∞ α) converges to an α-stable random variable in distribution, as t tends to infinity, where ψ α (t) = t 1 / α for 1 ≤ α ≤ 2 and ψ α (t) = t 2 − 1 α for 1 2 < α < 1. When θ < 0 , for all 1 2 < α ≤ 2 we show that, as t → ∞ , J t α (θ , ν , 0) := t e 1 2 θ t 2 X t α converges to ξ ∞ α − ν θ and J t α (θ , ν , n) : = − θ t 2 (J t α (θ , ν , n − 1) − (2 n − 3) ! ! (ξ ∞ α − ν θ)) → (2 n − 1) ! ! (ξ ∞ α − ν θ) a.s. for all n ≥ 1 , where (− 1) ! ! = 1 and ξ ∞ α = ∫ 0 ∞ s e 1 2 θ s 2 d M s α . [ABSTRACT FROM AUTHOR]

This paper presents asymptotic formulas in the case of the following two problems for the Pucci's extremal operators M ± . It is considered the solution u ε (x) of − ε 2 M ± ∇ 2 u ε + u ε = 0 in Ω such that u ε = 1 on Γ. Here, Ω ⊂ R N is a domain (not necessarily bounded) and Γ is its boundary. It is also considered v (x , t) the solution of v t − M ± ∇ 2 v = 0 in Ω × (0 , ∞) , v = 1 on Γ × (0 , ∞) and v = 0 on Ω × { 0 }. In the spirit of their previous works [Berti D, Magnanini R. Asymptotics for the resolvent equation associated to the game-theoretic p-laplacian. Appl Anal. 2019;98(10):1827–1842.; Berti D, Magnanini R. Short-time behavior for game-theoretic p-caloric functions. J Math Pures Appl (9). 2019;(126):249–272.], the authors establish the profiles as ϵ or t → 0 + of the values of u ε (x) and v (x , t) as well as of those of their q-means on balls touching Γ. The results represent a further step in the extensions of those obtained by Varadhan and by Magnanini-Sakaguchi in the linear regime. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behavior of the chemotaxis-Navier–Stokes system ∂ t n + u ⋅ ∇ n = λ Δ n − ∇ ⋅ (χ (c) n ∇ c) , ∂ t c + u ⋅ ∇ c = μ Δ c − f (c) n , ∂ t u + u ⋅ ∇ u + ∇ P = ζ Δ u − n ∇ φ , ∇ ⋅ u = 0 , t > 0 , x ∈ Ω posed in a strip domain Ω := R 2 × [ 0 , 1 ] ⊂ R 3 . In Peng-Xiang (Math. Models Methods Appl. Sci., 28 (2018), 869-920), the authors have established the global existence of strong solutions to this system with non-flux boundary conditions for n and c and non-slip boundary conditions for u. Our main purpose is to establish the time-decay rates for such solutions. This will be done by using the anisotropic L p interpolation and the iterative techniques. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following Choquard equation: − Δ u + u = (I α ∗ F (u)) F ′ (u) i n R N where N ⩾ 3 , α ∈ (0 , N) , I α is the Riesz potential, and F (u) := 1 p | u | p + 1 q | u | q , where p = N + α N and q = N + α N − 2 are lower and upper critical exponents in sense of the Hardy–Littlewood–Sobolev inequality. Based on perturbation method and the invariant sets of descending flow, we prove that the above equation possesses infinitely many sign-changing solutions. Our results extend the results in Seok [Nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2018;76:148–156] and Su [New result for nonlinear Choquard equations: doubly critical case. Appl Math Lett. 2020;102(106092):0–7]. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hénon–Hardy type systems on R n : (− Δ) α 2 u (x) = | x | a v p (x) , u (x) ≥ 0 , x ∈ R n , (− Δ) α 2 v (x) = | x | b u q (x) , v (x) ≥ 0 , x ∈ R n , where n ≥ 2 , n > α , 0 < α ≤ 2 or α = 2 m. We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on R n in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in R N . preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. ; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in R N and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hénon–Hardy type systems on R n : (− Δ) α 2 u (x) = | x | a v p (x) , u (x) ≥ 0 , x ∈ R n , (− Δ) α 2 v (x) = | x | b u q (x) , v (x) ≥ 0 , x ∈ R n , where n ≥ 2 , n > α , 0 < α ≤ 2 or α = 2 m. We prove Liouville theorems (i.e. non-existence of nontrivial nonnegative solutions) for the above Hénon–Hardy systems. The arguments used in our proof is the method of scaling spheres developed in [Dai W, Qin GLiouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752.]. Our results generalize the Liouville theorems for single Hénon–Hardy equation on R n in Bidaut-Véron and Pohozaev [Nonexistence results and estimates for some nonlinear elliptic problems. J Anal Math. 2001;84:1.49], Chen et al. [Liouville type theorems, a priori estimates and existence of solutions for critical order Hardy–Hénon equations in R N . preprint, submitted, arXiv: 1808.06609], Dai et al. [Liouville type theorems, a priori estimates and existence of solutions for non-critical higher-order Lane–Emden–Hardy equations. preprint, submitted for publication, arXiv: 1808–10771], Dai and Qin [Liouville type theorems for Hardy–Hénon equations with concave nonlinearities. Math Nachrichten. 2020;293(6):1084–1093. ; Liouville type theorems for fractional and higher-order Hénon–Hardy type equations via the method of scaling spheres. preprint, submitted for publication, arXiv: 1810.02752], Guo and Liu [Liouville-type theorems for polyharmonic equations in R N and in Liouville-type theorems for. Proc Roy Soc Edinburgh Sect A. 2008;138(2):339–359], and Phan and Souplet [Liouville-type theorems and bounds of solutions of Hardy–Hénon equations. J Diff Equ. 2012;252:2544–2562] to systems. [ABSTRACT FROM AUTHOR]