Abstract Motivated by Euclidean boxes, we consider “thin” annular domains of the form U = ( a , b ) × U 0 ⊆ R n $U=\left(a,b\right){\times}{U}_{0}\subseteq {\mathbb{R}}^{n}$ in polar coordinates, where the spherical base U 0 ⊆ S n − 1 ${U}_{0}\subseteq {\mathbb{S}}^{n-1}$ is an inner uniform domain. We show that, with respect to the measure φ U 2 ${\varphi }_{U}^{2}$ determined by the principal Dirichlet Laplacian eigenfunction φ U , such annular domains satisfy volume doubling and Poincaré inequalities uniformly over all locations and scales. This implies sharp Dirichlet heat kernel estimates expressed in terms of φ U . Our results hold uniformly over the collection of all annuli in R n ${\mathbb{R}}^{n}$ . We also give matching two-sided bounds for the first Dirichlet Laplacian eigenfunction and eigenvalue for some annular domains including annuli in R n ${\mathbb{R}}^{n}$ . Moreover, we prove eigenfunction inequalities for φ U under domain perturbations of U . The proofs of our main results utilize eigenfunction comparison techniques due to Lierl and the authors (“The Dirichlet heat kernel in inner uniform domains: Local results, compact domains and non-symmetric forms,” J. Funct. Anal. , vol. 266, no. 7, pp. 4189–4235, 2014; “Perturbing the principal Dirichlet eigenfunction,” arXiv preprint arXiv:2504.18783, 2025), small scale φ U 2 ${\varphi }_{U}^{2}$ -Poincaré inequalities (Lierl and Saloff-Coste, “The Dirichlet heat kernel in inner uniform domains: Local results, compact domains and non-symmetric forms,” J. Funct. Anal. , vol. 266, no. 7, pp. 4189–4235, 2014), as well as a discretization technique of Coulhon and Saloff-Coste (“Variétés riemanniennes isométriques à l’infini,” Rev. Mat. Iberoam. , vol. 11, pp. 687–726, 1995). Finally, our methods also imply uniform Neumann heat kernel estimates for thin annular domains.
Abstract On generalized Heisenberg-type groups G ( 2 n , m , U , W ) $\mathbb{G}\left(2n,m,\mathbb{U},\mathbb{W}\right)$ , we establish uniform volume estimates for the ball associated with a large class of Carnot-Carathéodory distances. As an application, we prove weak (1, 1) O ( C m n )-estimates for the corresponding centered Hardy-Littlewood maximal functions, extending the results in [C. Bi, H.-Q. Li, and Y. Zhang, “Centered Hardy-Littlewood maximal functions on H-type groups revisited,” Math. Ann. , vol. 391, no. 3, pp. 3765–3797, 2025]. As a by-product, we obtain uniformly volume doubling property on Heisenberg groups for a natural family of left-invariant Riemannian metrics.
Abstract Let G be a compact connected Lie group and P e , a ( G ) = C ([0, 1] → G | γ (0) = e , γ (1) = a ) be the pinned path space with a pinned Brownian motion measure ν λ , a defined by the heat kernel p ( λ −1 t , x , y ), where λ is a positive parameter. We consider a Witten Laplacian − L λ , D $-{L}_{\lambda ,\mathcal{D}}$ acting on functions with the Dirichlet boundary condition on a certain domain D ⊂ P e , a ( G ) $\mathcal{D}\subset {P}_{e,a}\left(G\right)$ which includes finitely many geodesics { l 1 , …, l N } between e and a . ν λ , a has the formal path integral expression ν λ , a ( d γ ) = Z λ − 1 exp − λ E ( γ ) d γ ${\nu }_{\lambda ,a}\left(\mathrm{d}\gamma \right)={Z}_{\lambda }^{-1}\mathrm{exp}\left(-\lambda E\left(\gamma \right)\right)\mathrm{d}\gamma $ , where E ( γ ) = 1 2 ∫ 0 1 | γ ̇ ( t ) | 2 d t $E\left(\gamma \right)=\frac{1}{2}{\int }_{0}^{1}\vert \dot {\gamma }\left(t\right){\vert }^{2}\mathrm{d}t$ and E is a Morse function when a is not a point of the cut-locus of e . Hence, by the analogy of finite dimensional cases, one may expect that the lowlying spectrum of − λ − 1 L λ , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$ can be approximated by the spectral sets of Ornstein-Uhlenbeck type operators which approximate − λ − 1 L λ , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$ in each small neighborhood of critical points { l i } when λ → ∞ . However, in contrast to the finite dimensional case, the spectral sets of the approximate Ornstein-Uhlenbeck type operators contain essential spectrum. It may be difficult to analyze the behavior of the spectrum of − λ − 1 L λ , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$ near the set of the essential spectrum. In this paper, we study the asymptotic behavior of the lowlying discrete spectrum of − λ − 1 L λ , D $-{\lambda }^{-1}{L}_{\lambda ,\mathcal{D}}$ in the complement of the neighborhood of the set of essential spectrum of the approximate Ornstein-Uhlenbeck type operators at { l i } as λ → ∞.
Let (A, tr) be a von Neumann algebra with a faithful, normal trace tr : A ->& Copf;. For each a is an element of A, define S(lambda, epsilon) = tr[log((a-lambda)(& lowast;)(a-lambda) + E)], lambda is an element of & Copf;, epsilon > 0, so that the limit as epsilon -> 0(+) of S is the log potential of the Brown measure of a. Suppose that for a fixed lambda is an element of & Copf;, the function epsilon bar right arrow partial derivative S/partial derivative epsilon (lambda, epsilon) = tr[log((a-lambda)(& lowast;)(a-lambda) + E)(-1)] admits a real analytic extension to a neighborhood of 0 in & Ropf;. Then we will show that lambda is outside the spectrum of a. We will apply this result to several examples involving circular and elliptic elements, as well as free multiplicative Brownian motions. In most cases, we will show that the spectrum of the relevant element a coincides with the support of its Brown measure.
This paper focuses on the global-in-time existence and singularity formation of smooth solutions for the pressure-gradient system with radial symmetry. The system owns one positive and one negative wave characteristics, which result that both expansion and convergence effects exist simultaneously for any initial data. We define the rarefaction and compression characters for the system and establish the a priori estimates of solutions by using the methods of characteristic decompositions and invariant regions. It is verified that the smooth solution exists globally for rarefaction initial data with vacuum at the origin, while singularities can develop in finite time for the initial data including strong compression somewhere. Moreover, the numerical simulation results verified the analysis in this paper.
We consider the existence of nontrivial solution u is an element of H-1 (& Ropf;(N) ) for the following nonlinear Choquard equation with an almost periodic term -Delta u + u = ( I-mu & lowast; (alpha( y)F(u))) alpha(x) f (u) in & Ropf;(N), where N >= 3, mu is an element of (0,N), I-mu is the Riesz potential, F is the primitive function of f , and alpha is almost periodic. As one will see, almost periodic functions of several variables lack some important features of periodic functions and almost periodic functions on & Ropf;. This makes our problem more intriguing and challenging.
In this paper, we consider weighted p-Laplacian equations with a gradient term and a nonlinear term Delta(p,f)u+(p-1)W(u)|del u|(p)+F(u)=0 on smooth metric measure spaces (M- n , g, e -f dV), where p > 1, W(t) is a continuous function for t > 0 and F(t) is a differentiable function in (0, infinity). By making some appropriate assumptions about W and F, we derive some Liouville-type theorems for positive solutions to the above equation, if (M n , g, e -f dV) satisfies Ric(f)(m)>= 0. . As applications we derive Liouville-type theorems of positive solutions to some generalized static Fisher-KPP equation, Allen-Cahn equation, static Newell-Whitehead equation and Lichnerowicz equation.
The paper deals with radially symmetric solutions of the following nonlinear Schrodinger-Poisson-Slater equation -Delta u +(|x|(-1) *u(2))u= mu|u|(p-2)u, in R-3, where (|x|(-1)* u(2)) is the repulsive Coulomb potential and mu > 0 is the Slater constant. When p = 18/5, we prove that the radially symmetric ground state solution of above equation is unbroken. Then, we obtain the uniqueness and smoothness of this positive radially symmetric solution and determine the expression for the best constant of the Coulomb-Sobolev inequality.
We investigate wave propagation in a bistable buffered system with nonlocal dispersal. A central question in this problem is the effect of stationary buffers on calcium waves. In this paper, we first establish the existence of bistable traveling waves, demonstrating that the buffers cannot eliminate the presence of calcium waves. We then investigate the range and sign of their speeds. We observe a new phenomenon: while the buffers do not alter the propagation direction of the calcium waves, they can reduce the speed of these waves. Finally, we establish the exact asymptotic behavior of the traveling waves at infinity and prove a form of stability for a pair of diverging traveling waves. Consequently, we derive a sufficient condition for solutions to converge to the larger stable equilibria.
Considered herein are the classification of solitary wave solutions of the degenerate rotation-two-component Camassa-Holm system. Depending on the values of the rotating parameter Omega and the balance index sigma, we classify various localized solitary wave solutions (smooth, peakons, and cuspons) for this model. We also prove that horizontally symmetric weak solution of this model must be traveling waves.
In this paper we consider the existence of novel multi-bumps solutions for the following Schrodinger equation (0.1) - Delta u + V(|x|)u = u(p), u > 0 in R-N , u is an element of H-1(R-N), where p is an element of (1 , N + 2/N - 2) and V is positive radial function satisfying V(|x| ) = V-0 + a/|x|(n) + O(1/|x|(n + theta)), as |x| -> infinity, with V-0, a, theta > 0 and n > 1. By introducing the Miranda theorem and developing some analytic techniques, combined with the Lyapunov-Schmit reduction method, we construct infinitely many new multi-bumps solutions of (0.1) when N >= 4. These multi-bump solutions possess a unique structure, with their bumps appearing in pairs along the third and fourth directions of x at spatial infinity, breaking through the conventional distribution patterns of solutions in previous studies. This result significantly complements and extends the previous research such as Duan and Musso ("New type of solutions for the nonlinear Schr & ouml;dinger equation in R-N," J. Differ. Equations, vol. 336, pp. 479-504, 2022), Guo et al. ("Non-degeneracy and existence of new solutions for the Schrodinger equations," J. Differ. Equations, vol. 326, pp. 254-279, 2022), Wei and Yan ("Infinitely many positive solutions for the nonlinear Schr & ouml;dinger equations in R-N," Calc. Var. Partial Differ. Equations, vol. 37, no. 3, pp. 423-439, 2010), further expanding the range from n > max {p - 1/4 , 2} to the case n > 1, bringing new perspectives and methodological expansions to the research in this field.
We study the Liouville equation triangle u + e(2u) = 0 in a Riemannian surface (M, g) with nonnegative Ricci curvature. Under some asymptotic lower bound assumptions, we classify all the solutions to this equation, meanwhile we obtain the rigidity results for the ambient manifold. Note that our assumptions are optimal in some sense and differ from the classical assumption of finite total curvature.
In this paper we introduce and study the nonlinear version of sampling Durrmeyer operators. For what concerns the space of continuous functions, a pointwise and uniform convergence theorem have been established. Additionally, approximation results for nonlinear sampling Durrmeyer operators in the general setting of Orlicz spaces are also provided. In particular, a general modular convergence theorem has been obtained via a density approach. As a corollary, it follows that this theory applies to particular cases of Orlicz spaces, such as to L p - spaces, interpolation spaces (L alpha log beta L - spaces) and exponential type spaces. Finally, various applications and examples, including graphical representations, are provided.
In this short paper, we are concerned with a nonlinear hyperbolic conservation system of three equations, which is an extended macroscopic model for traffic flow. The main contribution is to prove the global existence of weak solutions and obtain a new application of the compactness framework given in the paper "Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type" (J. Funct. Anal., 261(2011), 2797-2815).
This paper investigates the existence and multiplicity of solutions to the following double critical p-fractional Schrodinger-Poisson system with electromagnetic fields in & Ropf;(3) : {& varepsilon;(ps)(-Delta)(p,A & varepsilon;)(s)u + V(x)|u|(p-2)u - phi|u|(p)(s)& sharp;-2u = |u|(p)(s)*-2u + g(x,|u|(p))|u|(p-2)u in & Ropf;(3), (-Delta)(s)phi = |u|(p)(s)& sharp; n & Ropf;(3), where & varepsilon; > 0 is a small parameter, (-Delta)(p,A & varepsilon;)(s) denotes the p-fractional magnetic operator, with 3/4 < s < 1, 1 < p < 3/s, A is an element of C(& Ropf;(3), & Ropf;(3)) is a magnetic potential and A (& varepsilon;) (x): = & varepsilon;(-1)A(x), while p(s)* := 3p/3-sp and p(s)(& sharp;) := p(3+2s)/2(3-sp) are the Sobolev critical exponent and the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, respectively. The potential & Vscr; and nonlinearity g satisfy some natural structural conditions. The main features and novelty of the paper lie in the simultaneous appearance of the double critical exponents and the nonlocal terms, which require to control the relationship between the two critical exponents. In order to overcome the difficulties resulting from the double critical terms, we rely on the concentration-compactness principle and some detailed analysis. Moreover, existence and multiplicity of solutions for this system are obtained by a careful application of the variational method. In some way, the main conclusions of the paper extend and complement some results established in (B. L. Zhang, M. Squassina, and X. Zhang, "Fractional NLS equations with magnetic field, critical frequency and critical growth," Manuscripta Math., vol. 155, no. 1, pp. 115-140, 2018, Y. H. Ding and F. H. Lin, "Solutions of perturbed Schrodinger equations with critical nonlinearity," Calc. Var. Partial Differ. Equ., vol. 30, no. 2, pp. 231-249, 2007, X. He, "Positive solutions for Schrodinger-Poisson systems with doubly critical exponents," Appl. Math. Lett., vol. 120, 2021, Art. no. 107190, S. Qu and X. He, "Multiplicity of high energy solutions for fractional Schrodinger-Poisson systems with critical frequency," Electron. J. Differ. Equ., vol. 2022, no. 47, pp. 1-21, 2022, Y. Song and S. Shi, "Existence and multiplicity solutions for the p-fractional Schrodinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity," Complex Var. Elliptic Equ., vol. 64, no. 7, pp. 1163-1183, 2019, M. Yang, "Existence of semiclassical solutions for some critical Schrodinger-Poisson equations with potentials," Nonlinear Anal., vol. 198, 2020, Art. no. 111874, L. Zheng, "Solutions of perturbed fractional Schrodinger-Poisson system with critical nonlinearity in & Ropf;(3) ," Taiwan J. Math., vol. 25, no. 4, pp. 775-791, 2021).
This paper studies the number of limit cycles, known as the Smale-Pugh problem, for the generalized Abel equation dx/ d theta=A(theta)x(p)+B(theta)x(q), where A and B are piecewise trigonometrical polynomials of degree m with two zones 0 <= theta < theta (1) and theta (1) <= theta <= 2 pi. By means of the first and second order analysis using the Melnikov theory and applying the new Chebyshev criterion that established by J. Huang, H. Liang, and X. Zhang, "A new Chebyshev criterion and its application to planar differential systems," J. Differ. Equ., vol. 344, pp. 658-695, 2023, we estimate the maximum number of positive and negative limit cycles that such equations can have, and reveal how this maximum number, denoted by H-theta 1(m) , is affected by the location of the separation line theta = theta (1). For the equation of classical Abel type, our result not only includes the estimates provided in the recent paper (Huang et al., SIAM J. Appl. Dyn. Syst., 2020), i.e., H (2 pi) (m) >= 4m - 2 for theta (1) = 2 pi, but also shows that the equation in the discontinuous case can possess more than two times as many limit cycles as in the continuous case. More accurately, H (pi) (m) >= 8m + 2 and H-theta 1(m) >= 14m -6 for theta (1) is an element of (0, pi) boolean OR (pi, 2 pi).
This paper investigates the existence, uniqueness, and asymptotic behavior of pullback measure attractors and evolution systems of measures for a complex-valued p-Laplacian Ginzburg–Landau lattice system (GLLS) driven by superlinear Lévy noise. We first derive several long-time uniform estimates and tail-end estimates of the solutions, and establish the existence and uniqueness of pullback measure attractors for the non-autonomous dynamical system generated by the solution operators in the space of probability measures. We then study the existence of evolution systems of measures and analyze their limiting behavior. The upper semicontinuity of the pullback measure attractors is also discussed. Under a large damping assumption, we prove that the pullback measure attractor reduces to a singleton, which yields the existence, uniqueness and exponential mixing of periodic/invariant measures and evolution systems of measures for the corresponding stochastic systems. A key result shows that the limit of a sequence of evolution systems of measures of the p-Laplacian GLLS with superlinear Lévy noise must be an evolution system of measures of the corresponding limiting system. The difficulties caused by the superlinear noise coefficients, the nonlinear p-Laplace operator, the lack of compactness in infinite-dimensional lattices, and the inapplicability of Fatou’s lemma in the non-autonomous case are overcome by using the polynomial dissipation of the drift term, the uniform tail-end estimates, the large-time pullback argument, and the monotonicity-balance conditions. Finally, some numerical simulations are presented to illustrate the behavior of the stochastic GLLS driven by superlinear Lévy noise.
In this paper, we study the following critical Hartree system {-epsilon(2)Delta u= V-1(X)u=& micro;(1)(|x|(-4)*|u|(2))u+beta(|x|(-4)*|v(2)|u, x is an element of R (N)(,) -epsilon(2)Delta u= V-2(X)u=& micro;(2)(|x|(-4)*|u|(2))u+beta(|x|(-4)*|v(2)|u, x is an element of R (N), where N >= 5,V-1(x),V-2(x) is an element of L-N/(2)(& Ropf;(N)) boolean AND L-loc(infinity) (& Ropf;(N)) are non-negative and of critical frequency,epsilon>0 is a small parameter,mu 1,mu 2,beta are positive constants. By using the Lusternik-Schnirelmann category theory and varia-tional methods, we relate the number of semi-classical states with the topology of the zero set Omega=V-1(-1)(0)boolean AND V-2(-1)(0) for epsilon small enough.
This paper is devoted to quantitative refinements of the nonlocal Sobolev inequality with explicit determination of stability constants. In the single-bubble regime, we rigorously prove that the optimal stability constant c(HLS) = inf (N)(u is an element of D1,2(& Ropf;) )\M ( & Vert;del u & Vert;(2) (2)(L)(& Ropf;(N))-S-HLS .f(& Ropf;)(N) (I-mu & lowast; |u|(2 & lowast;)(mu) ) |u|(2 & lowast;)(mu) dx )1/(2 & lowast;)(mu)/ (distD1,2)(& Ropf;(N))(u, M) is strictly smaller than the spectral gap constant associated with sequences approaching the manifold Al of non-local Sobolev optimizers. Here S-HLS denotes the optimal Hardy-Littlewood-Sobolev constant. The proof relies on a precise asymptotic expansion of the nonlocal Sobolev quotient along a suitably chosen family of test functions converging to M. Furthermore, we establish a strict upper bound for c(HLS)in the multi-bubble regime, showing that c(HLS)
In this paper we obtain uniformly locally L^∞-estimate of solutions to non-autonomous quasilinear system involving operators in divergence form and a family of nonlinearities that are allowed to grow also critically.
