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In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in R N . Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: ] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities). [ABSTRACT FROM AUTHOR]

We study the nonlinear coupled Kirchhoff system with purely Sobolev critical exponent. By using appropriate transformation, we get one equivalent system involving a critical Schrödinger system and an algebraic system. Through solving the critical Schrödinger system with a corresponding algebraic system, under suitable conditions we obtain the existence and classification of positive ground states for the Kirchhoff system in dimensions 3 and 4. Furthermore, for the degenerate case, we give a complete classification of positive ground states for the Kirchhoff system in any dimension. To the best of our knowledge, this paper is the first to give classification results for the ground states of Kirchhoff systems. The results in this paper partially extend and complement the main results established by Lü and Peng [Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differ. Equ. 263 (2017) 8947–8978] considering the linearly coupled Kirchhoff system with subcritical exponent and some partial results established by Chen and Zou [Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012) 515–551; Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differ. Equ. 52 (2015) 423–467], where the authors considered the coupled purely critical Schrödinger system.

In this paper, we consider the multiplicity and concentration phenomenon of positive solutions to the following Choquard equation - ε 2 Δ u + V (x) u = ε - α Q (x) (I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ - 2 u + f (u) in R N , where N ≥ 3 , (N - 4) + < α < N , I α is the Riesz potential, ε is a small parameter, V (x) ∈ C (R N) ∩ L ∞ (R N) is a positive potential, f ∈ C 1 (R + , R) is a subcritical nonlinear term and 2 α ∗ = N + α N - 2 is the upper-critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. By means of variational methods and delicate energy estimates, we establish the relationship between the number of solutions and the profiles of potentials V and Q, and the concentration behavior of positive solutions is also obtained for ε > 0 small. [ABSTRACT FROM AUTHOR]

This paper studies the nonlocal fractional analog of the famous paper of Brezis and Nirenberg [Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477]. Namely, we focus on the following model: P−Δsu−λu=α|u|p−2u+β|u|2s∗−2uinΩ,u=0inRN∖Ω,$$\begin{align*}\hskip5pc {\left(\mathcal{P}\right)} {\left\{ \def\eqcellsep{&}\begin{array}{l} {\left(-\Delta \right)}^s u-\lambda u = \alpha |u|^{p-2}u + \beta |u|^{2^*_s-2}u \quad \mbox{in}\quad \Omega ,\\ u=0\quad \mbox{in}\quad \mathbb {R}^N\setminus \Omega , \end{array} \right.}\hskip-5pc \end{align*}$$where (−Δ)s$(-\Delta)^s$ is the fractional Laplace operator, s∈(0,1)$s \in (0,1)$, with N>2s$N > 2s$, 20,λ,α∈R$\beta >0,\, \lambda , \alpha \in \mathbb {R}$, and establish the existence of nontrivial solutions and sign‐changing solutions for the problem (P)$(\mathcal{P})$. [ABSTRACT FROM AUTHOR]

The (bitwise) complement x of a binary word x is obtained by changing each 0 in x to 1 and vice versa. An antisquare is a nonempty word of the form x x. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is (5 +√5)/2. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is good if the only antisquares it contains are 01 and 10. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length n and determine the repetition threshold between polynomial and exponential growth for the number of good words. [ABSTRACT FROM AUTHOR]

We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete. [ABSTRACT FROM AUTHOR]

This paper dealt with a class of Kirchhoff type equations involving singular nonlinearity in a closed Riemannian manifold $ (M, g) $ of dimension $ n\ge3 $. Existence and uniqueness of a positive weak solution were obtained under certain assumptions with the help of the variation methods and some analysis techniques.

The Kirchhoff equation was proposed in 1883 by Kirchhoff [Vorlesungen über Mechanik, Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and PDE’s, G. M. de la Penha, and L. A. Medeiros, Eds., Amsterdam, North-Holland, 1978] returned to the equation and proposed a general Kirchhoff equation in arbitrary dimension with external force term which was written as ∂2u∂t2+a+b∫Ω|∇u|2dxΔu=f(x,u), $\frac{{\partial }^{2}u}{\partial {t}^{2}}+\left(a+b{\int }_{{\Omega}}\vert \nabla u{\vert }^{2}\mathrm{d}x\right){\Delta}u=f\left(x,u\right),$ where Δ=−∑∂2∂xi2 ${\Delta}=-\sum \frac{{\partial }^{2}}{\partial {x}_{i}^{2}}$ is the Laplace-Beltrami Euclidean Laplacian. We investigate in this paper a closely related stationary version of this equation, in the case of closed manifolds, when u is vector valued and when f is a pure critical power nonlinearity. We look for the stability of the equations we consider, a question which, in modern nonlinear elliptic PDE theory, has its roots in the seminal work of Gidas and Spruck.

In this paper, we consider the Cauchy problem of a time-fractional nonlinear diffusion equation. According to Kaplan’s first eigenvalue method, we first prove the blow-up of the solutions in finite time under some sufficient conditions. We next provide sufficient conditions for the existence of global solutions by using the results of Zhang and Sun. In conclusion, we find the second critical exponent for the existence of global and non-global solutions via the decay rates of the initial data at spatial infinity.

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: −M∫Ω∇updxΔpu=up∗−2u+λup−2u−up−2ulnu2 x∈Ω, u=0 x∈∂Ω, where Ω⊂ℝN is a bounded domain with a smooth boundary, 2

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

In this paper we investigate the critical exponents of two families of Pucci's extremal operators. The notion of critical exponent that we have chosen for these fully nonlinear operators which are not variational is that of threshold between existence and nonexistence of the solutions for semilinear equations with pure power nonlinearities. Interesting new exponents appear in this context. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the first eigenvalue of p-Laplacian and the relation between the first eigenvalue and the existence (or nonexistence) of nontrivial (positive) solution for quasilinear elliptic obstacle problems. Utilizing the fact that obstacle problem have consanguineous relations with corresponding equation, we get a simple approach to study the properties of solutions of obstacle problems, such as existence and nonexistence, regularity and stability, etc. In this paper we are mainly concerned with the existence and nonexistence. [Copyright &y& Elsevier]

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

This paper considers blow-up and global existence for a semilinear space-time fractional pseudo-parabolic equation with nonlinear memory in a bounded domain. We determine the critical exponents of the Cauchy problem when $ \alpha < \gamma $ and $ \alpha\ge \gamma, $ respectively. The results obtained in this study are noteworthy extension to the results of time-fractional differential equation. The critical exponent is consistent with the corresponding Cauchy problem for the time-fractional differential equation with nonlinear memory, which illustrates that the diffusion effect of the third order term is not strong enough to change the critical exponents.

In this paper, we consider a class of Lane–Emden heat flow system with the fractional Laplacian \begin{equation*} \begin{cases} u_{t} +(-\Delta)^\frac{\alpha}{2}u = N_{1}(v)+f_{1}(x), & \ (x,t) \in Q,\\ v_{t}+(-\Delta)^\frac{\alpha}{2}v = N_{2}(u)+f_{2}(x), & \ (x,t)\in Q,\\ u(x,0)=a(x), v(x,0)=b(x), & \ x \in \mathbb R^{N}, \end{cases} \end{equation*} where $0

This paper is devoted to an initial-boundary value problem for a class of $ p $-Kirchhoff type parabolic equations. Firstly, we consider this problem with a general nonlocal coefficient $ M(\|\nabla u\|_p^p) $ and a general nonlinearity $ k(t)f(u) $. A new finite time blow-up criterion is established, also, the upper and lower bounds for the blow-up time are derived. Secondly, we deal with the case that $ M(\|\nabla u\|_p^p) = a+b\|\nabla u\|_p^p $, $ k(t)\equiv1 $ and $ f(u) = |u|^{q-1}u $, which was considered by Li and Han [Math. Model. Anal. 2019; 24: 195-217] only for $ q > 2p-1 $. The threshold results for the existence of global and finite time blow-up solutions to this problem are obtained for the case $ 1 < q\leq 2p-1 $, which, together with the results given by Li and Han, shows that $ q = 2p-1 $ is critical for the existence of finite time blow-up solutions to this problem. These results partially generalize and extend some recent ones in previous literature.

In this paper, we consider a class of critical Schrödinger-Bopp-Podolsky system. By virtue of the Nehari manifold and variational methods, we study the existence, nonexistence and asymptotic behavior of ground state solutions for this problem.

We study the normalized solutions for nonlinear Kirchhoff equation with Sobolev critical exponent in high dimensions $ \mathbb{R}^N(N\geqslant4) $. In particular, in dimension $ N = 4 $, there is a special phenomenon for Kirchhoff equation that the mass critical exponent $ 2+\frac{8}{N} $ is equal to the energy critical exponent $ \frac{2N}{N-2} $, which leads to the fact that the equation no longer has a variational structure in dimensions $ N\geqslant 4 $ if we consider the mass supercritical case, and remains unsolved in the existing literature. In this paper, by using appropriate transform, we first get the equivalent system of Kirchhoff equation. With the equivalence result, we obtain the nonexistence, existence and multiplicity of normalized solutions by variational methods, Cardano's formulas and Pohožaev identity.

In this paper, we study the following Schrödinger-Poisson system with critical exponent where $1 < p < 2$ and $\lambda > 0.$ Under suitable conditions on $k$, $h$ and $s$, we show that there exists $\lambda^{\ast}>0$ such that the above problem possesses infinitely many solutions with negative energy for each $\lambda\in(0, \lambda^{\ast})$. Moreover, we prove the existence of infinitely many solutions with positive energy. The main tools are the concentration compactness principle, $Z_{2}$ index theory and Fountain Theorem. These results extend some existing results in the literature.

Abstract: In this paper, a semilinear system of elliptic equations is investigated, which involves multiple critical exponents and singular points. By variational methods and analytic techniques, the related best Hardy–Sobolev constants is studied and the existence of positive solutions is established. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a nonlocal diffusion equation with Dirichlet boundary conditions and a reaction term Our main result shows that this nonlocal diffusion problem has a critical exponent which coincides with that of the corresponding local diffusion problem. The blow-up rate of solutions is also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, we study the long-time behavior of plate equation with nonlinear damping and critical nonlinearity. We prove the existence of a global attractor in the space . [Copyright &y& Elsevier]

Abstract: In this paper, we consider dynamical behavior of non-autonomous plate-type evolutionary equations with critical nonlinearity. We prove the existence of a uniform attractor in the space . [Copyright &y& Elsevier]

Abstract: In this paper, we prove the existence of a pullback attractor for a nonautonomous wave equation with critical exponent. To do this, we first use the concept of pullback -asymptotic compactness given in [T. Caraballo, G. Łukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (2006) 484–498] to show the existence of pullback attractor of nonautonomous dynamical systems, and then we give a new method for verifying the pullback -asymptotic compactness. [Copyright &y& Elsevier]

Abstract: In this paper we make the analysis of the blow up of low energy sign-changing solutions of a semilinear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence and nonexistence of global positive solutions for two evolution P-Laplacian equations in exterior domains with inhomogeneous boundary conditions. We demonstrate that is its critical exponent provided . Furthermore, we prove that if , then every positive solution of the equations blows up in finite time; whereas for , the equations admit the global positive solutions for some boundary value and some initial data . We also demonstrate that every positive solution of the equations blows up in finite time provided . [Copyright &y& Elsevier]

Abstract: In this paper we continue the analysis of the blow-up of low energy sign-changing solutions of semi-linear elliptic equations with critical Sobolev exponent, started in [M. Ben Ayed, K. El Mehdi, F. Pacella, Blow-up and nonexistence of sign-changing solutions to the Brezis–Nirenberg problem in dimension three, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. In addition we prove axial symmetry results for the same kind of solutions in a ball. [Copyright &y& Elsevier]

Abstract: In this paper we study low energy sign changing solutions of the critical exponent problem in Ω, on ∂Ω, where Ω is a smooth bounded domain in and λ is a real positive parameter. We make a precise blow-up analysis of this kind of solutions and prove some comparison results among some limit values of the parameter λ which are related to the existence of positive or of sign changing solutions. [Copyright &y& Elsevier]

In this paper, we deal with a class of fractional Hénon equation and by using the Lyapunov-Schmidt reduction method, under some suitable assumptions, we derive the existence of infinitely many solutions, whose energy can be made arbitrarily large. Compared to the previous works, we encounter some new challenges because of the nonlocal property for fractional Laplacian. But by doing some delicate estimates for the nonlocal term we overcome the difficulty and find infinitely many nonradial solutions.

In this paper, we consider a time fractional diffusion system with a nonlinear memory term in a bounded domain. We mainly prove some blow-up and global existence results for this problem. Moreover, we also give the decay estimates of the global solutions. Our proof relies on the eigenfunction method combined with the asymptotic behavior of the solution of a fractional differential inequality system, the estimates of the solution operators and the asymptotic behavior of the Mittag–Leffler function. In particular, we give the critical exponents of this problem in different cases. Our results show that, in some cases, whether one of the initial values is identically equal to zero has a great influence on blow-up and global existence of the solutions for this problem, which is a remarkable property of time fractional diffusion systems because the classical diffusion systems can not admit this property.

In this paper, we study a class of quasilinear elliptic equations with $\Phi$-Laplacian operator and critical growth. Using the symmetric mountain pass theorem and the concentration-compactness principle, we demonstrate that there exists $\lambda_i>0$ such that our problem admits $i$ pairs of nontrivial weak solutions provided $\lambda\in (0, \lambda_i)$.

This paper is concerned with the following quasilinear Schrödinger equations with critical exponent: \begin{equation*}\label{eqS0.1} - \Delta _p u+ V(x)|u|^{p-2}u - \Delta _p(|u|^{2\omega}) |u|^{2\omega-2}u = a k(x)|u|^{q-2}u+b |u|^{2\omega p^{*}-2}u,\qquad x\in\mathbb{R}^N. \end{equation*} Here $\Delta _p u =\mathrm{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator with $1< p < N$, $p^* =\frac{Np}{N-p}$ is the critical Sobolev exponent. $1\le 2\omega

In this paper, we discuss the existence and uniqueness of solutions for a non-autonomous reaction-diffusion equation with delay, after we prove the existence of a pullback 𝒟-asymptotically compact process. By a priori estimates, we show that it has a pullback 𝒟-absorbing set that allow us to prove the existence of a pullback 𝒟-attractor for the associated process to the problem.

We study a critical exponent of the flocking behavior to the one-dimensional 1D Cucker–Smale (C–S) model with a regular inverse power law communication on a general network with a spanning tree. For this, we propose a new nonlinear functional which can control the velocity diameter and decays exponentially fast as time goes on. As an application of the time-evolution of the nonlinear functional, we show that the C–S model on a line exhibits a unique critical exponent for unconditional flocking on a general network so that this improves an earlier result [S.-Y. Ha and J.-G. Liu, A simple proof of Cucker–Smale flocking dynamics and mean field limit, Commun. Math. Sci.7 (2009) 297–325.] on the all-to-all network. Our result also resolves the critical exponent conjecture posed in Cucker–Dong's work [On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci.19 (2009) 1391–1404] for 1D setting. Emergent behavior of the C–S model is independent of the special structure of the underlying network, as long as it contains a spanning tree. [ABSTRACT FROM AUTHOR]

The paper is concerned with the slightly subcritical elliptic problem with Hardy-critical term

In this paper, by using the concentration--compactness principle of Lions for variable exponents and variational arguments, we obtain the existence and multiplicity solutions for a class of $p(x)$-Kirchhoff type equations with critical exponent.

The self-avoiding walk, and lattice spin systems such as the ϕ4 model, are models of interest both in mathematics and in physics. Many of their important mathematical problems remain unsolved, particularly those involving critical exponents. We survey some of these problems, and report on recent advances in their mathematical understanding via a rigorous non-perturbative renormalization group method. [ABSTRACT FROM AUTHOR]

In this paper, we study the following problems {Δ2u−M(∫Ω|∇u|2dx)Δu=λf(x,u)+|u|2∗−2uinΩu=Δu=0on∂Ω, where 2∗=2NN−4 is the critical exponent. Under some conditions on M and f, we prove the existence of nontrivial solutions by using variational methods.