Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions,local means and non‐smooth atomic decompositions

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.     

Semi‐classical resolvent estimates and regions free of resonances

We prove resolvent estimates for self‐adjoint operators of the form on , , where is a semi‐classical parameter and , , is a real‐valued potential. The potential is supposed to have very little regularity with respect to the radial variable, only. As a consequence, we obtain a region free of resonances in the case when V is of compact support.     

On a generalized nonlinear heat equation in Besov and Triebel–Lizorkin spaces

The paper deals with solutions of the Cauchy problem of a nonlinear generalized heat equation in the context of Besov and Triebel–Lizorkin spaces and where and with initial data belonging to some spaces where and .     

Almost minimizers for semilinear free boundary problems with variable coefficients

We study regularity results for almost minimizers of the functional where is a matrix with Hölder continuous coefficients. In the case we show that an almost minimizer belongs to , where the exponent η is related with the competition between the Hölder continuity of the matrix , the parameter of almost minimization and γ. In some sense, this regularity is optimal. As far as the case is concerned, our results show that an almost minimizer is locally in each phase and , improving in some sense a recent result of David & Toro.     

Uniqueness results for semilinear elliptic systems on

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form As an application we consider the nonlinear Schrödinger system for and exponents q which satisfy in case and in case . Generalizing the results of Wei and Yao for we find new sufficient conditions and necessary conditions on such that precisely one positive solution exists. Our results dealing with the special case are optimal. Finally, an application to a multi‐component nonlinear Schrödinger system is given.     

Low dimensional instability for quasilinear problems of weighted exponential nonlinearity

We prove a sharp Liouville type theorem for stable solutions of the equation on the entire Euclidean space , where and f is a continuous and nonnegative function in such that as , where and . Our theorem holds true for and is sharp in the case .     

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.     

Semiclassical ground state solutions for a Schrödinger equation in with critical exponential growth

Consider a Schrödinger equation where and are two continuous real functions on , ε is a positive parameter, the nonlinearity f is assumed to be of critical exponential growth in the sense of the Trudinger‐Moser inequality. By truncating the potentials and , we are able to establish some new existence and concentration results for critical Schrödinger equation in by variational methods. As a particular case, we observe that the concentration appears at the maximum set of the nonlinear potential which complements the results in 6 , 23 .     

The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

This paper gives an insight into making a mathematical bridge between the parabolic‐parabolic signal‐dependent chemotaxis system and its parabolic‐elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic‐parabolic chemotaxis system with strong signal sensitivity to that for the parabolic‐elliptic chemotaxis system where Ω is a bounded domain in () with smooth boundary, is a constant and χ is a function generalizing In chemotaxis systems parabolic‐elliptic systems often gave some guide to methods and results for parabolic‐parabolic systems. However, the relation between parabolic‐elliptic systems and parabolic‐parabolic systems has not been studied except for the case that . Namely, in the case that Ω is a bounded domain, it still remains to analyze on the following question: Does a solution of the parabolic‐parabolic system converge to that of the parabolic‐elliptic system as ? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.     

On unique continuation for Schrödinger operators of fractional and higher orders

In this note we study the property of unique continuation for solutions of , where V is in a function class of potentials including for . In particular, when , our result gives a unique continuation theorem for the fractional Schrödinger operator in the full range of α values.     

Positive solutions of Kirchhoff type elliptic equations in with critical growth

In this paper, we study the following Kirchhoff type elliptic problem with critical growth: where , and , and the nonlinear growth term reaches the Sobolev critical exponent since for four spatial dimensions. In a non‐radial symmetric function space, we establish a local compactness splitting lemma of critical version to investigate the existence of positive ground state solutions.     

Spectral gaps for the Dirichlet‐Laplacian in a 3‐D waveguide periodically perturbed by a family of concentrated masses

We consider a spectral problem for the Laplace operator in a periodic waveguide perturbed by a family of “heavy concentrated masses”; namely, Π contains small regions of high density, which are periodically distributed along the z axis. Each domain has a diameter and the density takes the value in and 1 outside; m and ε are positive parameters, , . Considering a Dirichlet boundary condition, we study the band‐gap structure of the essential spectrum of the corresponding operator as . We provide information on the width of the first bands and find asymptotic formulas for the localization of the possible gaps.     

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

The main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in . More precisely, let and denote then we will prove in this article that there exists such that can be achieved for all , .     

Well‐posedness,blow‐up phenomena and analyticity for a two‐component higher order Camassa–Holm system

In this paper, the local well‐posedness for the Cauchy problem of a two‐component higher‐order Camassa–Holm system (2HOCH) is established in Besov spaces with and (and also in Sobolev spaces with ), which improves the corresponding results for higher‐order Camassa–Holm in 7 , 24 , 25 , where the Sobolev index is required, respectively. Then the precise blow‐up mechanism and global existence for the strong solutions of 2HOCH are determined in the lowest Sobolev space with . Finally, the Gevrey regularity and analyticity of the 2HOCH are presented.     

On analyticity of the ‐Stokes semigroup for some non‐Helmholtz domains

Consider the Stokes equations in a sector‐like C ³ domain . It is shown that the Stokes operator generates an analytic semigroup in for . This includes domains where the ‐Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in and L ² by constructing a suitable non‐Helmholtz projection to solenoidal spaces.     

Positive solutions of nonlinear Schrödinger equation with peaks on a Clifford torus

We prove the existence of large energy positive solutions for a stationary nonlinear Schrödinger equation with peaks on a Clifford type torus. Here where with for all Each is a function and is defined by the generalized notion of spherical coordinates. The solutions are obtained by a or a process.     

Some characterizations of BLO space

For , a real‐valued function belongs to space if In this paper, we establish a version of John–Nirenberg inequality suitable for the space with . As a corollary, it is proved that spaces are independent of the scale in sense of norm. Also, we characterize the space through weighted Lebesgue spaces and variable Lebesgue spaces, respectively.     

Maximal regularity for non‐autonomous Robin boundary conditions

We consider a non‐autonomous Cauchy problem where is associated with the form , where V and H are Hilbert spaces such that V is continuously and densely embedded in H. We prove H‐maximal regularity, i.e., the weak solution u is actually in (if and ) under a new regularity condition on the form with respect to time; namely Hölder continuity with values in an interpolation space. This result is best suited to treat Robin boundary conditions. The maximal regularity allows one to use fixed point arguments to some non linear parabolic problems with Robin boundary conditions.     

Variational and Perron–Wiener solutions in stratified Lie groups

We consider sub‐Laplacians in stratified Lie groups and we compare Perron–Wiener and weak‐variational solutions of the Dirichlet problem , where Ω is a bounded open set in and φ is the restriction to the boundary of a function such that . The result we obtained extends a previous theorem by Arendt and Daners, related to the classial Laplacian in .     

Sobolev estimates for (pseudo)‐differential operators and applications

In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.