Abstract In this article, we investigate the existence of periodic solutions to McKean–Vlasov stochastic differential equations subject to periodic Lyapunov conditions with distributional dependence. The proof is based on the construction of periodic Markov processes on the product space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mi mathvariant="script">P</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> ${\mathbb{R}}^{n}{\times}\mathcal{P}\left({\mathbb{R}}^{n}\right)$ , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">P</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> $\mathcal{P}\left({\mathbb{R}}^{n}\right)$ is the space of probability measures on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> ${\mathbb{R}}^{n}$ . Moreover, we also prove the existence of periodic solutions under Lyapunov conditions, where the Lyapunov functions involve only the spatial components. To illustrate our analysis, we present several concrete examples.

Abstract In this study, we explore the positive solutions of a nonlinear Choquard equation involving the Green kernel of the fractional operator ( − Δ B N ) − α ⁄ 2 {\left(-{\Delta }_{{{\mathbb{B}}}^{N}})}^{-\alpha /2} in the hyperbolic space, where Δ B N {\Delta }_{{{\mathbb{B}}}^{N}} represents the Laplace-Beltrami operator on B N {{\mathbb{B}}}^{N} , with α ∈ ( 0 , N ) \alpha \in \left(0,N) and N ≥ 3 N\ge 3 . This study is analogous to the Choquard equation in the Euclidean space, which involves the nonlocal Riesz potential operator. We consider the functional setting within the Sobolev space H 1 ( B N ) {H}^{1}\left({{\mathbb{B}}}^{N}) , employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to the fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the problem ( F α , λ {F}_{\alpha ,\lambda } ) in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.

Abstract In this article, we consider a free boundary problem modeling the growth of a double-layered tumor, which contains quiescent cells and proliferating cells. Compared with that composed of necrotic cells and proliferating cells, the remarkable feature of this model is that quiescent cells actively consume nutrients, introducing an additional nutrient decay term, but at a rate different from that of proliferating cells. The parameter μ \mu in model represents the “aggressiveness” of the tumor. A threshold μ * {\mu }_{* } is determined such that the radially symmetric stationary solution is linearly stable for μ &lt; μ * \mu \lt {\mu }_{* } and unstable for μ &gt; μ * \mu \gt {\mu }_{* } under non-radially symmetric perturbations. Moreover, it is shown that there exist a sequence { μ n } ( ≥ μ * ) \left\{{\mu }_{n}\right\}\left(\ge {\mu }_{* }) and a positive integer n * * ≥ 2 {n}^{* * }\ge 2 , such that a branch of symmetry-breaking stationary solutions bifurcates from the radially symmetric stationary solution for μ = μ n ( even n ≥ n * * ) \mu ={\mu }_{n}\left({\rm{even}}\hspace{0.33em}n\ge {n}^{* * }) .

Abstract In this work, we develop the classical theory of monotone and pseudomonotone operators in the class of convex-constrained Dirichlet-type problems involving fractional Riesz gradients in bounded and in unbounded domains Ω ⊂ R d \Omega \subset {{\mathbb{R}}}^{d} . We consider the problem of finding u ∈ K s u\in {K}^{s} , such that ∫ R d a ( x , u , D s u ) ⋅ D s ( v − u ) d x + ∫ Ω b ( x , u , D s u ) ( v − u ) d x ≥ 0 \mathop{\int }\limits_{{{\mathbb{R}}}^{d}}{\boldsymbol{a}}\left(x,u,{D}^{s}u)\cdot {D}^{s}\left(v-u){\rm{d}}x+\mathop{\int }\limits_{\Omega }b\left(x,u,{D}^{s}u)\left(v-u){\rm{d}}x\ge 0 for all v ∈ K s v\in {K}^{s} . Here, K s ⊂ Λ 0 s , p ( Ω ) {K}^{s}\subset {\Lambda }_{0}^{s,p}\left(\Omega ) is a non-empty, closed, and convex set of a fractional Sobolev-type space Λ 0 s , p ( Ω ) {\Lambda }_{0}^{s,p}\left(\Omega ) with 0 ≤ s ≤ 1 0\le s\le 1 and 1 &lt; p &lt; ∞ 1\lt p\lt \infty , and D s {D}^{s} denote the distributional Riesz fractional gradient, with two limit cases: D 1 = D {D}^{1}=D represents the classical gradient in the classical Sobolev space Λ 0 1 , p ( Ω ) = W 0 1 , p ( Ω ) {\Lambda }_{0}^{1,p}\left(\Omega )={W}_{0}^{1,p}\left(\Omega ) , and − D 0 = R -{D}^{0}=R denotes the vector-valued Riesz transform within Λ 0 0 , p ( Ω ) = { u ∈ L p ( R d ) : u = 0 a.e. in R d \ Ω } {\Lambda }_{0}^{0,p}\left(\Omega )=\left\{u\in {L}^{p}\left({{\mathbb{R}}}^{d}):\hspace{0.33em}u\left=0\hspace{0.33em}\hspace{0.1em}\text{a.e. in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{d}\backslash \Omega \right\} . We discuss the existence and uniqueness of solutions in this novel framework, and we obtain new results on the continuous dependence, with respect to the fractional parameter s s , of variational solutions corresponding to several classical assumptions on the structural functions a {\boldsymbol{a}} and b b adapted to the fractional framework. We introduce an extension of the Mosco convergence for convex sets K s {K}^{s} with respect to the parameter s s , including the limit cases s = 1 s=1 and s = 0 s=0 , to prove weak or strong convergences of the solutions u s {u}_{s} and their fractional gradients D s u {D}^{s}u , according to different cases. Several applications are illustrated with examples of unilateral problems, including quasi-variational inequalities with constraints of obstacle type u ≥ ψ u\ge \psi and s s -gradient type ∣ D s u ∣ ≤ g | {D}^{s}u| \le g .

Abstract This article verifies the low Mach number and non-resistive limit of local strong solutions to non-isentropic compressible magnetohydrodynamic (MHD) equations in general three-dimensional bounded domains when the temperature variation is large but finite. The uniform estimates of strong solutions are obtained in a short time interval independent of the Mach number and the magnetic resistivity coefficient, provided that the initial data are well prepared. Previous results on this multi-scale singular limit of compressible MHD equations are either for the cases of the weak solutions or for the cases of small temperature variations or for the cases of flat boundary where the mechanisms are essentially different. The new ingredient of this article is that we develop a new approach of weighted energy to establish the uniform estimates for high-order spatial derivatives of solutions.

Abstract By using critical point theory in combination with periodic approximations, we obtain novel sufficient conditions for the existence of nontrivial homoclinic solutions for a class of periodic partial difference equations with sign-changing mixed nonlinearities. Notably, in certain specific cases, a necessary and sufficient condition is also established.

Abstract In this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function ϕ : [ 0 , ∞ ) → [ 0 , ∞ ) \phi :\left[0,\infty )\to \left[0,\infty ) . Given any bounded domain with the slice property, the authors prove that it is a generalized Hajłasz-Sobolev embedding domain if and only if it is a generalized F F -weak cigar domain, where F F is a modulus of continuity related to the weight function ϕ \phi of the generalized Hajłasz-Sobolev spaces under consideration. Comparing with the classical Hajłasz-Soblev spaces, one main difficulty in dealing with generalized Hajłasz-Sobolev spaces lies in that both its smoothness weight function ϕ \phi and the related modulus of continuity F F have no explicit expression. To overcome this, the authors introduce and use some key indices to accurately describe the increasing or the decreasing behavior of both ϕ \phi and F F . Besides the classical Hajłasz-Sobolev spaces, this result can be applied to many other nontrivial spaces such as Hajłasz-Sobolev spaces with logarithmic smoothness and is of wide generality.

Abstract In this article, we consider the following double critical fractional Schrödinger-Poisson system involving p -Laplacian in R 3 {{\mathbb{R}}}^{3} of the form: ε s p ( − Δ ) p s u + V ( x ) ∣ u ∣ p − 2 u − ϕ ∣ u ∣ p s ♯ − 2 u = ∣ u ∣ p s * − 2 u + f ( u ) in R 3 , ε s p ( − Δ ) s ϕ = ∣ u ∣ p s ♯ in R 3 , \left\{\begin{array}{l}{\varepsilon }^{sp}{\left(-\Delta )}_{p}^{s}u+{\mathcal{V}}\left(x){| u| }^{p-2}u-\phi {| u| }^{{p}_{s}^{\sharp }-2}u={| u| }^{{p}_{s}^{* }-2}u+f\left(u)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ {\varepsilon }^{sp}{\left(-\Delta )}^{s}\phi ={| u| }^{{p}_{s}^{\sharp }}\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right. where ε &gt; 0 \varepsilon \gt 0 is a positive parameter, s ∈ ( 3 4 , 1 ) s\in \left(\frac{3}{4},1) , ( − Δ ) p s {\left(-\Delta )}_{p}^{s} is the fractional p -Laplacian operator, p ∈ ( 3 2 , 3 ) p\in \left(\frac{3}{2},3) , p s * = 3 p 3 − s p {p}_{s}^{* }=\frac{3p}{3-sp} is the Sobolev critical exponent, p s ♯ = p 2 ( 3 + 2 s ) ( 3 − s p ) {p}_{s}^{\sharp }=\frac{p}{2}\frac{\left(3+2s)}{\left(3-sp)} is the upper exponent in the sense of the Hardy-Littlewood-Sobolev inequality, V ( x ) : R 3 → R {\mathcal{V}}\left(x):{{\mathbb{R}}}^{3}\to {\mathbb{R}} symbolizes a continuous potential function with a local minimum, and the continuous function f f possesses subcritical growth. With the help of well-known penalization methods and Ljusternik-Schnirelmann category theory, we use the topological arguments to attain the multiplicity and concentration of the positive solutions for the above system.

Abstract In this article, we investigate the initial value problem for the hyperbolic geometric flow equation, which is derived from hyperbolic geometric flow for Riemannian metric. We find that some of the quadratic nonlinear terms satisfy the null condition by rearranging the equation. Based on the ghost weight and standard energy estimate, the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> </m:math> {L}^{1}-{L}^{\infty } estimate of solutions, and the bootstrap argument, lifespan of smooth solutions is established under the assumption of small data.

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