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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]

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]

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]

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]

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]

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]