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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]