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In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of rational numbers satisfies a certain numerical condition. However there remains a question how to obtain explicit descriptions of the values. In this paper, we give a method to do this, which is a further development of the technique in [4]., Comment: 22pages, To appear in Communications in Contemporary Mathematics

We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual “pillar-type” domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper.

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space ℍ4, ∫ℍ4 e32π2u2 − 1 (1 + |u|)2 dvg ≤ C∥u∥L2(ℍ4)2, (0.1) for all u ∈ Cc∞(ℍ4) with ∫ ℍ4(P2u)udvg ≤ 1. We will also establish an Adachi–Tanaka-type inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

In this paper, we consider a free boundary model in one space dimension which describes the spreading of a species subject to climate change, where favorable environment is shifting away with a constant speed [Formula: see text] and replaced by a deteriorated yet still favorable environment. We obtain two threshold speeds [Formula: see text] and a complete classification of the long-time dynamics of the model, which reveals significant differences between the cases [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. For example, when [Formula: see text], for a suitably parameterized family of initial functions [Formula: see text] increasing continuously in [Formula: see text], we show that there exists [Formula: see text] such that the species vanishes eventually when [Formula: see text], it spreads with asymptotic speed [Formula: see text] when [Formula: see text], it spreads with forced speed [Formula: see text] when [Formula: see text], and it spreads with speed [Formula: see text] when [Formula: see text]. Moreover, in the last case, while the spreading front propagates with asymptotic speed [Formula: see text], the profile of the population density function [Formula: see text] approaches a propagating pair consisting of a traveling wave with speed [Formula: see text] and a semi-wave with speed [Formula: see text].

This paper deals with a nonlocal diffusion elliptic eigenvalue problem. Specifically, the diffusion of the unknown variable at a point of the domain depends on its value in a neighborhood of the point. We apply bifurcation arguments and appropriate approximation to obtain our results. Some applications to the population dynamics will be given.

In this paper, we consider the following critical equation: [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if [Formula: see text], [Formula: see text] has a stable critical point [Formula: see text] with [Formula: see text] and [Formula: see text], then the above equation has infinitely many positive solutions, where [Formula: see text] is the unique positive solution of [Formula: see text] with [Formula: see text]. Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606–2633], it implies that the role of stable critical points of [Formula: see text] in constructing bump solutions is more important than that of [Formula: see text] and that [Formula: see text] can influence the sign of [Formula: see text], i.e. [Formula: see text] can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of [Formula: see text] which include the case of a saddle point.

We construct Ionel–Parker’s proposed refinement of the standard relative Gromov–Witten invariants in terms of abelian covers of the symplectic divisor and discuss in what sense it gives rise to invariants. We use it to obtain some vanishing results for the standard relative Gromov–Witten invariants. In a separate paper, we describe to what extent this refinement sharpens the usual symplectic sum formula and give further qualitative applications.

In this paper, we study the existence and concentration behaviors of positive solutions to the following fractional nonlinear Schrödinger equation: [Formula: see text] where [Formula: see text] is a positive parameter, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is the fractional Laplacian. When the potential [Formula: see text] decays to zero like [Formula: see text], [Formula: see text], and [Formula: see text] like [Formula: see text] with [Formula: see text], we will show that the existence of ground states [Formula: see text] belonging to [Formula: see text], which concentrates at a minimum point of the auxiliary function [Formula: see text].

In this paper, we study the local well-posedness for the Cauchy problem of a semilinear fractional diffusion equation where the perturbations behave like [Formula: see text] and [Formula: see text], and [Formula: see text] is the characteristic function of a ball [Formula: see text]. Here, we are interested in the solvability of the problem when singular initial data [Formula: see text] are taken in [Formula: see text]. Eventually, we give sufficient conditions to the nonexistence of positive global solutions.

In this paper, we sharpen the lower bound on the codimension of the irreducible components of the Noether–Lefschetz locus of surfaces in projective toric threefolds given in [U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1–22]. We also provide a simpler proof of Theorem 4.11 in [U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1–22], which allows one to avoid some technical assumptions.

International audience; The subject matter of this paper concerns the derivation of fluid limits for gyro-kinetic models. The arguments apply for any collision kernel satisfying the usual conservations (mass, momentum, kinetic energy) and possessing a production entropy sign. We describe the set of equilibria in terms of several moments, we determine the average collision invariants, and we write the associated macro-scopic equations and the entropy inequality.

Let [Formula: see text] be a connected finite graph and [Formula: see text] be the set of functions defined on [Formula: see text]. Let [Formula: see text] be the discrete [Formula: see text]-Laplacian on [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is positive everywhere. Consider the operator [Formula: see text]. We prove that [Formula: see text] is one-to-one, onto and preserves order. So it implies that there exists a unique solution to the equation [Formula: see text] for any given [Formula: see text]. We also prove that the equation [Formula: see text] has a solution which is unique up to a constant, where [Formula: see text] is the average of [Formula: see text]. With the help of these results, we finally give various conditions such that the [Formula: see text]th Kazdan–Warner equation [Formula: see text] has a solution on [Formula: see text] for given [Formula: see text] and [Formula: see text]. Thus we generalize Grigor’yan, Lin and Yang’s work Kazdan–Warner equation on graph, Calc. Var. Partial Differential Equations 55(4) (2016) Paper No. 92, 13 pp. for [Formula: see text] to any [Formula: see text].

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

This paper develops the notion and properties of the generalized principal eigenvalue for an elliptic system coupling an equation in a plane with one on a line in this plane, together with boundary conditions that express exchanges taking place between the plane and the line. This study is motivated by the reaction–diffusion model introduced by Berestycki, Roquejoffre and Rossi [The influence of a line with fast diffusion on Fisher–KPP propagation, J. Math. Biol. 66(4–5) (2013) 743–766] to describe the effect on biological invasions of networks with fast diffusion imbedded in a field. Here we study the eigenvalue associated with heterogeneous generalizations of this model. In a forthcoming work [Influence of a line with fast diffusion on an ecological niche, preprint (2018)] we show that persistence or extinction of the associated nonlinear evolution equation is fully accounted for by this generalized eigenvalue. A key element in the proofs is a new Harnack inequality that we establish for these systems and which is of independent interest.

In this paper, we investigate spreading properties of the solutions of the Kolmogorov-Petrovsky-Piskunov-type, (to be simple,KPP-type) lattice system \begin{equation}\label{firstequation}\overset{.}u_{i}(t) =d^{\prime}_{i}(u_{i+1}(t)-u_{i}(t))+d_{i}(u_{i-1}(t)-u_{i}(t))+f(i,u_{i}).\end{equation} we develop some new discrete Harnack-type estimates and homogenization techniques for this lattice system to construct two speeds $\overline\omega \leq \underline \omega$ such that $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{i\geq\omega t}|u_i(t)|=0$ for any $\omega>\overline{\omega}$, and $\displaystyle{\lim_{t\rightarrow+\infty}}\sup \limits_{0\leq i\leq\omega t}|u_i(t)-1|=0$ for any $\omega

In this paper, we propose a combinatorial approach to study Calabi–Yau threefolds constructed as a resolution of singularities of a double covering of [Formula: see text] branched along an arrangement of eight planes. We use this description to give a complete classification of arrangements of eight planes in [Formula: see text] defining Calabi–Yau threefolds modulo projective transformation with [Formula: see text] and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).

In this paper, we study some type of equations which may model the behavior of species inhabiting in some habitat. For our purpose, using a priori bounded techniques, we obtain a positive solution to a family of non-local partial differential equations with non-homogeneous boundary conditions.

In this paper, the minimization problem [Formula: see text] where [Formula: see text] is studied when [Formula: see text] ([Formula: see text]) is an open, bounded, convex domain with smooth boundary and [Formula: see text]. We show that [Formula: see text] is either zero, when the maximum of the distance function to the boundary of [Formula: see text] is greater than [Formula: see text], or it is a positive real number, when the maximum of the distance function to the boundary of [Formula: see text] belongs to the interval [Formula: see text]. In the latter case, we provide estimates for [Formula: see text] and show that for [Formula: see text] sufficiently large [Formula: see text] coincides with the principal frequency of the [Formula: see text]-Laplacian in [Formula: see text]. Some particular cases and related problems are also discussed.

This paper not only presents a criterion for the Cordes–Nirenberg space [Formula: see text] imbedding between the associate Morrey space [Formula: see text] and the Morrey space [Formula: see text], but also characterizes a Radon measure [Formula: see text] such that the Cordes–Nirenberg potential space [Formula: see text] is restricted to the [Formula: see text]-based Campanato space [Formula: see text] — thereby giving an application of the discovered characterization to the regularity for a class of the elliptic equations with symmetric [Formula: see text]-coefficients.

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.

In this paper, existence and multiplicity of positive solutions of the elliptic system − Δu + V1(x)u = μ1(x)u3 + β(x)uv2in Ω 𝜀, − Δv + V2(x)v = β(x)u2v + μ 2(x)v3 in Ω 𝜀,u = v = 0 on ∂Ω𝜀 is proved, where Ω𝜀 is an exterior domain in ℝN such that ℝN∖Ω 𝜀 is far away from the origin and contains a sufficiently large ball, N = 1, 2, 3, and the coefficients Vj,μj,β are continuous functions on ℝN which tend to positive constants at infinity. We do not assume μ1,μ2,β to be positive functions.

This paper concerns a global controllability problem for heat equations. In the absence of control, the solution to the linear heat system globally exists. While for each initial data, we can find a feedback control acting on an internal subset of the space domain such that the corresponding solution to the system blows up at given time.

In this paper, we derive the following Leray–Trudinger type inequality on a bounded domain [Formula: see text] in [Formula: see text] containing the origin. [Formula: see text] [Formula: see text] Here, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text] for [Formula: see text] This improves an earlier result by Psaradakis and Spector. Also, we prove that for any [Formula: see text] in the place of [Formula: see text], the above inequality is false if we take [Formula: see text].

In this paper, we study an evolution equation involving the normalized [Formula: see text]-Laplacian and a bounded continuous source term. The normalized [Formula: see text]-Laplacian is in non-divergence form and arises for example from stochastic tug-of-war games with noise. We prove local [Formula: see text] regularity for the spatial gradient of the viscosity solutions. The proof is based on an improvement of flatness and proceeds by iteration.

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

In this paper, we characterize a family of solitary waves for nonlinear Schrödinger equation (NLS) with derivative (DNLS) by the structure analysis and the variational argument. Since DNLS does not enjoy the Galilean invariance any more, the structure analysis here is closely related with the nontrivial momentum and shows the equivalence of nontrivial solutions between the quasilinear and the semilinear equations. Firstly, for the subcritical parameters [Formula: see text] and the critical parameters [Formula: see text], we show the existence and uniqueness of the solitary waves for DNLS, up to the phase rotation and spatial translation symmetries. Secondly, for the critical parameters [Formula: see text], [Formula: see text] and the supercritical parameters [Formula: see text], there is no nontrivial solitary wave for DNLS. At last, we make use of the invariant sets, which is related to the variational characterization of the solitary wave, to obtain the global existence of solution for DNLS with initial data in the invariant set [Formula: see text], with [Formula: see text], [Formula: see text] or [Formula: see text]. On the one hand, different with the scattering result for the [Formula: see text]-critical NLS in [B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math. 285(5) (2015) 1589–1618], the scattering result of DNLS does not hold for initial data in [Formula: see text] because of the existence of infinity many small solitary/traveling waves in [Formula: see text] with [Formula: see text], [Formula: see text] or [Formula: see text]. On the other hand, our global result improves the global result in [Y. Wu, Global well-posedness of the derivative nonlinear Schrödinger equations in energy space, Anal. Partial Differential Equations 6(8) (2013) 1989–2002; Global well-posedness on the derivative nonlinear Schrödinger equation, Anal. Partial Differential Equations 8(5) (2015) 1101–1112] (see Corollary 1.6).

In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] is a nonnegative potential function with nonempty zero set [Formula: see text], [Formula: see text] is a positive function such that the operator [Formula: see text] is indefinite and non-degenerate for [Formula: see text] large. We show that both in subcritical and critical cases, equation [Formula: see text] admits a least energy solution which for [Formula: see text] large localized near the zero set [Formula: see text].

Let [Formula: see text] and [Formula: see text] be the local Hardy space in the sense of D. Goldberg. In this paper, the authors establish two bilinear decompositions of the product spaces of [Formula: see text] and their dual spaces. More precisely, the authors prove that [Formula: see text] and, for any [Formula: see text], [Formula: see text], where [Formula: see text] denotes the local BMO space, [Formula: see text], for any [Formula: see text] and [Formula: see text], the inhomogeneous Lipschitz space and [Formula: see text] a variant of the local Orlicz–Hardy space related to the Orlicz function [Formula: see text] for any [Formula: see text] which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.

In this paper, we study the existence of positive multi-peak solutions to the fractional Schrödinger–Poisson system [Formula: see text] where [Formula: see text] is a small parameter, [Formula: see text] is a positive function, [Formula: see text] and [Formula: see text] Under some given conditions which are given in Sec. ??, we prove the existence of a positive solution with m-peaks and concentrating near a given local maximum point of [Formula: see text]

In this paper, we consider the following singular elliptic problem involving an exponential nonlinearity in two dimensions: [Formula: see text] [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. We show the existence and multiplicity of positive solutions globally with respect to the bifurcation parameter [Formula: see text].

金沢大学人間社会研究域学校教育系, In this paper, the authors characterize the Sobolev spaces (Formula presented.) with (Formula presented.) and (Formula presented.) via a generalized Lusin area function and its corresponding Littlewood–Paley (Formula presented.)-function. The range (Formula presented.) is also proved to be nearly sharp in the sense that these new characterizations are not true when (Formula presented.) and (Formula presented.). Moreover, in the endpoint case (Formula presented.), the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces. © 2017 World Scientific Publishing Company, Embargo Period 12 months

In this paper we will prove that suitable weak solutions of three dimensional Navier-Stokes equations in bounded domain can be constructed by a particular type of artificial compressibility approximation.

We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate., Comment: 21 pages. Communications on contemporary mathematics, in press

In this paper, we prove the global gradient estimates on the generalized Lebesgue spaces for weak solutions to elliptic quasilinear obstacle problems. It is worth noticing that the coefficients related to the obstacle problems are merely measurable with small BMO norms and the underlying domain does not satisfy any smoothness conditions.

In this paper, we prove the local existence and uniqueness of the solution with discontinuous density for the inhomogeneous non-resistive magnetohydrodynamics (MHD) equations on a [Formula: see text] bounded domain [Formula: see text] or [Formula: see text], if the initial data [Formula: see text] with [Formula: see text] satisfies [Formula: see text].

In this paper, the existence and multiplicity of solutions for a quasilinear elliptic problem driven by the [Formula: see text]-Laplacian operator is established. These solutions are also built as ground state solutions using the Nehari method. The main difficulty arises from the fact that the [Formula: see text]-Laplacian operator is not homogeneous and the nonlinear term is indefinite.

This paper is devoted to investigate the symmetry and monotonicity properties for positive solutions of fractional Laplacian equations. Especially, we consider the following fractional Laplacian equation with homogeneous Dirichlet condition: [Formula: see text] Here [Formula: see text] is a domain (bounded or unbounded) in [Formula: see text] which is convex in [Formula: see text]-direction. [Formula: see text] is the nonlocal fractional Laplacian operator which is defined as [Formula: see text] Under various conditions on [Formula: see text] and on a solution [Formula: see text] it is shown that [Formula: see text] is strictly increasing in [Formula: see text] in the left half of [Formula: see text], or in [Formula: see text]. Symmetry (in [Formula: see text]) of some solutions is proved.

In this paper, by Morse theory, we study the existence and multiplicity of solutions for the [Formula: see text]-Laplacian equation with a “concave” nonlinearity and a parameter. In our results, we do not need any additional global condition on the nonlinearities, except for a subcritical growth condition.

Let [Formula: see text], [Formula: see text]. Then we have the algebraic curve [Formula: see text], and its coordinate algebras (the Riemann surfaces) [Formula: see text] and [Formula: see text] The Lie algebras [Formula: see text] and [Formula: see text] are called the [Formula: see text]th superelliptic Lie algebras associated to [Formula: see text]. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form [Formula: see text].

In this paper, we are concerned with the zero Dirichlet boundary value problem associated to the quasilinear elliptic equation [Formula: see text] where [Formula: see text] is an open and bounded set in [Formula: see text] ([Formula: see text]), [Formula: see text] is a continuously differentiable real function in [Formula: see text], [Formula: see text] is an elliptic, bounded and symmetric matrix, [Formula: see text] is non-negative and may be singular at zero and [Formula: see text]. We give sufficient conditions on [Formula: see text], [Formula: see text] and [Formula: see text] in order to have a comparison principle and, as a consequence, uniqueness of positive solutions being continuous up to the boundary.

In this paper, we study [Formula: see text]-expansive homeomorphisms on surfaces. We prove that when [Formula: see text] is a [Formula: see text]-expansive homeomorphism defined on a compact boundaryless surface [Formula: see text] with non-wandering set [Formula: see text] being the whole of [Formula: see text] then [Formula: see text] is expansive. This condition on the non-wandering set cannot be relaxed: we present an example of a [Formula: see text]-expansive homeomorphisms on a surface with genus [Formula: see text] with wandering points that is not expansive.

In this paper we extend the well known theorem of Angelo Lopez concerning the Picard group of the general space projective surface containing a given smooth projective curve, to the intermediate N\'eron-Severi group of a general hypersurface in any smooth projective variety., Comment: 14 pages, to appear on Communications in Contemporary Mathematics

To characterize the complete structure of the Fučík spectrum of the [Formula: see text]-Laplacian on higher dimensional domains is a long-standing problem. In this paper, we study the [Formula: see text]-Laplacian with integrable potentials on an interval under the Dirichlet or the Neumann boundary conditions. Based on the strong continuity and continuous differentiability of solutions in potentials, we will give a comprehensive characterization of the corresponding Fučík spectra: each of them is composed of two trivial lines and a double-sequence of differentiable, strictly decreasing, hyperbolic-like curves; all asymptotic lines of these spectral curves are precisely described by using eigenvalues of the [Formula: see text]-Laplacian with potentials; and moreover, all these spectral curves have strong continuity in potentials, i.e. as potentials vary in the weak topology, these spectral curves are continuously dependent on potentials in a certain sense.

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $\Sigma$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $\Sigma$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers., Comment: in Communication in Contemporary Mathematics, 2015

The paper studies the existence of an exponential attractor for the wave equation with structural damping and supercritical nonlinearity [Formula: see text]. By constructing a bounded absorbing set with higher global regularity (rather than the long-standing partial regularity) and by using the weak quasi-stability estimates (rather than the strong ones as usual), we establish the existence of an exponential attractor in the natural energy space.

This paper establishes such a uniqueness principle that under [Formula: see text] the fractional order differential inequality [Formula: see text] has the property that if [Formula: see text] then a non-negative weak solution to [Formula: see text] is unique, and if [Formula: see text] then the uniqueness of a non-negative weak solution to [Formula: see text] occurs when and only when [Formula: see text], thereby innovatively generalizing Gidas–Spruck’s result for [Formula: see text] in [Formula: see text] discovered in [B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981) 525–598].

This paper concerns solutions with time-dependent singularities for a semilinear parabolic equation with a superlinear absorption term. Here, by time-dependent singularity, we mean a singularity with respect to the space variable whose position depends on time. It is shown that if the power of the nonlinearity is in some range, then any singularity is removable. On the other hand, in other range, two types of time-dependent singular solutions exist: One resembles the fundamental solution of the Laplace equation near the singular point, and the other has a stronger singularity.

In this paper we study the optimization problem $$\mu^\star(\Omega):=\inf_{u\in \semi}\frac{\into |\n u|^2 \dx}{\into V u^2 \dx}$$ in a suitable functional space $\semi$. Here, $V$ is the multi-singular potential given by $$V:=\sum_{1\leq i\mu^\star(\rr^N)$ when $n\geq 3$ and $\mu^\star(\Omega)=\mu^\star(\rr^N)$ when $n=2$ (It is known from \cite{cristi1} that $\mu^\star(\rr^N)=(N-2)^2/n^2)$. In the situation when all the poles are located on the boundary we show that $\mu^\star(\Omega)=N^2/n^2$ if $\Omega$ is either a ball, the exterior of a ball or a half-space. Our results do not depend on the distances between the poles. In addition, in the case of boundary singularities we obtain that $\mu^\star(\Omega)$ is attained in $\hoi$ when $\Omega$ is a ball and $n\geq 3$. Besides, $\mu^\star(\Omega)$ is attained in $\semi$ when $\Omega$ is the exterior of a ball with $N\geq 3$ and $n\geq 3$ whereas in the case of a half-space $\mu^\star(\Omega)$ is attained in $\semi$ when $n\geq 3$. We also analyze the critical constants in the so-called \textit{weak} Hardy inequality which characterizes the range of $\mu's$ ensuring the existence of a lower bound for the spectrum of the Schr\"{o}dinger operator $-\Delta -\mu V$. In the context of both interior and boundary singularities we show that the critical constants in the weak Hardy inequality are $(N-2)^2/(4n-4)$ and $N^2/(4n-4)$, respectively., Comment: revisions

In this paper, we study a class of Trudinger–Moser inequality in the Sobolev space [Formula: see text]. Setting [Formula: see text] we prove: [Formula: see text] [Formula: see text] for [Formula: see text] for [Formula: see text], and [Formula: see text] there exist extremal functions for [Formula: see text] if [Formula: see text]. Blow-up analysis, elliptic estimates and a version of compactness result due to Lions are used to prove (1) and (3). The proof of (2) is based on computations of testing functions which are a combination of eigenfunctions with the Moser sequence.

Given a continuous dynamical system [Formula: see text] on a compact metric space [Formula: see text] and a continuous potential [Formula: see text], the generalized rotation set is the subset of [Formula: see text] consisting of all integrals of [Formula: see text] with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measure-theoretic entropies over all invariant measures whose integrals produce that point. In this paper, we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at a particular exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy.