Spectral heat content for Lévy processes

In this paper we study the spectral heat content for various Lévy processes. We establish the small time asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in , . We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in with respect to symmetric Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally, we establish that the small time asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.     

Gradient estimates of Dirichlet heat kernels for unimodal Lévy processes

Under some mild assumptions on the Lévy measure and the symbol we obtain gradient estimates of Dirichlet heat kernels for pure‐jump isotropic unimodal Lévy processes in .     

On quasi‐infinitely divisible distributions with a point mass

An infinitely divisible distribution on is a probability measure μ such that the characteristic function has a Lévy–Khintchine representation with characteristic triplet , where ν is a Lévy measure, and . A natural extension of such distributions are quasi‐infinitely distributions. Instead of a Lévy measure, we assume that ν is a “signed Lévy measure”, for further information on the definition see [10]. We show that a distribution with and , where is the absolutely continuous part, is quasi‐infinitely divisible if and only if for every . We apply this to show that certain variance mixtures of mean zero normal distributions are quasi‐infinitely divisible distributions, and we give an example of a quasi‐infinitely divisible distribution that is not continuous but has infinite quasi‐Lévy measure. Furthermore, it is shown that replacing the signed Lévy measure by a seemingly more general complex Lévy measure does not lead to new distributions. Last but not least it is proven that the class of quasi‐infinitely divisible distributions is not open, but path‐connected in the space of probability measures with the Prokhorov metric.     

Exponential moments of first passage times and related quantities for Lévy processes

For a Lévy process on the real line, we provide complete criteria for the finiteness of exponential moments of the first passage time into the interval , the sojourn time in the interval , and the last exit time from . Moreover, whenever these quantities are finite, we derive their respective asymptotic behavior as .     

Solutions with sign information for nonlinear nonhomogeneous problems

We consider a parametric elliptic equation driven by a nonlinear nonhomogeneous differential operator and with a Robin boundary condition. In the first part we prove the existence of positive solutions and state a bifurcation‐type result describing how the set of positive solutions changes as the parameter varies. In the second part we show that problem admits nodal (sign‐changing) solutions provided the parameter is sufficiently large.     

Singular Hamiltonian elliptic systems with critical exponential growth in dimension two

We will focus on the existence of nontrivial solutions to the following Hamiltonian elliptic system where are numbers belonging to the interval [0, 2), V is a continuous potential bounded below on by a positive constant and the functions f and g possess exponential growth range established by Trudinger–Moser inequalities in Lorentz–Sobolev spaces. The proof involves linking theorem and a finite‐dimensional approximation.     

Estimates of densities for Lévy processes with lower intensity of large jumps

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times.     

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

Let M be a Riemannian manifold with a smooth boundary. The main question we address in this article is: “When is the Laplace–Beltrami operator , , invertible?” We consider also the case of mixed boundary conditions. The study of this main question leads us to the class of manifolds with boundary and bounded geometry introduced by Schick (Math. Nachr. 223 (2001), 103–120). We thus begin with some needed results on the geometry of manifolds with boundary and bounded geometry. Let be an open and closed subset of the boundary of M. We say that has finite width if, by definition, M is a manifold with boundary and bounded geometry such that the distance from a point to is bounded uniformly in x (and hence, in particular, intersects all connected components of M). For manifolds with finite width, we prove a Poincaré inequality for functions vanishing on , thus generalizing an important result of Sakurai (Osaka J. Math, 2017). The Poincaré inequality then leads, as in the classical case to results on the spectrum of Δ with domain given by mixed boundary conditions, in particular, Δ is invertible for manifolds with finite width. The bounded geometry assumption then allows us to prove the well‐posedness of the Poisson problem with mixed boundary conditions in the higher Sobolev spaces , .     

Multiple solutions for a class of quasilinear Schrödinger equations

In this paper, we study the following quasilinear Schrödinger equations of the form where , , . Some existence results for positive solutions, negative solutions and sequence of high energy solutions are obtained via a perturbation method.     

Spectral statistics of random Schrödinger operator with growing potential

In this work we investigate the spectral statistics of random Schrödinger operators acting on where are i.i.d random variables distributed uniformly on [0,1].     

On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth

In this paper we study the existence of solutions for fractional Schrödinger equations of the form where V is a potential bounded and the nonlinear term has the critical exponential growth. We prove the existence of at least one weak solution by combining the mountain‐pass theorem with the Trudinger–Moser inequality and a version of a result due to Lions for critical growth in .     

Multiple solutions for Kirchhoff‐type problems with variable exponent and nonhomogeneous Neumann conditions

The existence of at least three weak solutions for a class of differential equations with ‐Kirchhoff‐type and subject to small perturbations of nonhomogeneous Neumann conditions is established under suitable assumptions. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.     

Weighted estimates for the multilinear maximal function on the upper half‐spaces

For a general dyadic grid, we give a Calderón–Zygmund type decomposition, which is the principle fact about the multilinear maximal function on the upper half‐spaces. Using the decomposition, we study the boundedness of . We obtain a natural extension to the multilinear setting of Muckenhoupt's weak‐type characterization. We also partially obtain characterizations of Muckenhoupt's strong‐type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen–Pérez type weighted estimates.     

Non‐autonomous forms and invariance

We generalize the Beurling–Deny–Ouhabaz criterion for parabolic evolution equations governed by forms to the non‐autonomous, non‐homogeneous and semilinear case. Let be Hilbert spaces such that V is continuously and densely embedded in H and let be the operator associated with a bounded H‐elliptic form for all . Suppose is closed and convex and the orthogonal projection onto . Given and , we investigate when the solution of the non‐autonomous evolutionary problem remains in and show that this is the case if for a.e. . Moreover, we examine necessity of this condition and apply this result to a semilinear problem.     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Existence and asymptotic behaviour for the time‐fractional Keller–Segel model for chemotaxis

One of the most important systems for understanding chemotactic aggregation is the Keller–Segel system. We consider the time‐fractional Keller–Segel system of order . We prove an existence result with small initial data in a class of Besov–Morrey spaces. Self‐similar solutions are obtained and we also show an asymptotic behaviour result.     

On a ‐Kirchhoff equation with critical exponent and an additional nonlocal term via truncation argument

We study existence and multiplicity of solutions of the following nonlocal ‐Kirchhoff equation with critical exponent, via truncation argument on the Sobolev space with variable exponent, where Ω is a bounded smooth domain of , , M, f are continuous functions, , and are real parameter.     

The concentration behavior of ground state solutions for a critical fractional Schrödinger–Poisson system

In this paper, we study the following critical fractional Schrödinger–Poisson system where is a small parameter, and , is the fractional critical exponent for 3‐dimension, has a positive global minimum, and are positive and have global maximums. We obtain the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials and Q as the concentration position of these ground state solutions as . Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.     

Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

We study solutions of the problem (0.1) where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D₀ and D₁ are starshaped with respect to the same point , then the same occurs for every superlevel set of u.     

Fractional Kirchhoff problem with critical indefinite nonlinearity

We study the existence and multiplicity of positive solutions for a family of fractional Kirchhoff equations with critical nonlinearity of the form where is a smooth bounded domain, and . Here M is the Kirchhoff coefficient and is the fractional critical Sobolev exponent. The parameter λ is positive and the is a real valued continuous function which is allowed to change sign. By using a variational approach based on the idea of Nehari manifold technique, we combine effects of a sublinear and a superlinear term to prove our main results.