We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in ℝ^N Δ^2u+V(ε x)u=λu+μ|u|^q-2u+|u|^2^**-2u ℝ^N, ∫_ℝ^N|u|^2dx=c^2, where Δ^2 is the biharmonic operator, N≥5, μ,c>0, ε>0, λ∈ℝ, q∈(2,2+8/N), and 2^**=2N/N-4 is the Sobolev critical exponent. The potential V is a bounded and continuous nonnegative function, satisfying some suitable global conditions. Using minimization techniques and a truncation argument, we show that the number of normalized solutions is not less than the number of global minimum points of V when the parameter ε is sufficiently small. To overcome the loss of compactness of the energy functional due to the critical growth, we apply the concentration-compactness principle. To the best of our knowledge, this study is the first contribution regarding the concentration and multiplicity properties of normalized solutions of critical biharmonic equations with combined nonlinearities in ℝ^N. To some extent, the main results included in this paper complement several recent contributions to the study of biharmonic equations with combined nonlinearities.

This study is devoted to proving the existence of weak solutions for a nonlinear elliptic problem with Neumann-type boundary data. The problem is driven by a discontinuous power nonlinearity and a nonsmooth prescribed data. Additionally, we aim to derive an estimate that proves the well-posedness of the problem. This estimate serves as an evidence for the uniqueness of the existing solution when the boundary term is “smooth".

This article is devoted to the interplay between productively Menger and productively Hurewicz subspaces of the Cantor space. In particular, we show that in the Laver model for the consistency of the Borel's conjecture these two notions coincide and characterize Hurewicz spaces. On the other hand, it is consistent with CH that there are productively Hurewicz subspaces of the Cantor space which are not productively Menger. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

In this chapter we give a geometric representation of H_n(B;𝕃) classes, where 𝕃 is the 4-periodic surgery spectrum, by establishing a relationship between the normal cobordism classes 𝒩^H_n(B,∂) and the n-th 𝕃-homology of B, representing the elements of H_n(B;𝕃) by normal degree one maps with a reference map to B. More precisely, we prove that for every n ≥ 6 and every finite complex B, there exists a map Γ: H_n(B;𝕃) ⟶𝒩^H_n(B,∂).

This paper deals with the following fractional (p,q)-Choquard equation with exponential growth of the form: epsilon(ps)(-Delta)(p)(s)u+epsilon(qs)(-Delta)(q)(s)u+Z(x)(|u|(p-2)u+|u|(q-2)u) = epsilon(mu-N)[|x|(-mu) * F(u)]f(u)in R-N, where s is an element of (0, 1), epsilon > 0 is a parameter, 2 <= p = N/s < q, and 0 < mu < N. The nonlinear function f has an exponential growth at infinity and the continuous potential function Z satisfies suitable natural conditions. With the help of the Ljusternik-Schnirelmann category theory and variational methods, the multiplicity and concentration of positive solutions are obtained for epsilon > 0 small enough. In a certain sense, we generalize some previously known results.

This article is devoted to the interplay between productively Menger and productively Hurewicz subspaces of the Cantor space. In particular, we show that in the Laver model for the consistency of the Borel's conjecture these two notions coincide and characterize Hurewicz spaces. On the other hand, it is consistent with CH that there are productively Hurewicz subspaces of the Cantor space which are not productively Menger.

The aim of this paper is to study existence results for a singular problem involving the p-biharmonic operator and the Hardy potential. More precisely, by combining monotonicity arguments with the variational method, the existence of solutions is established. By using the Nehari manifold method, the multiplicity of solutions is proved. An example is also given to illustrate the importance of these results.

In this paper, existence of pairs of solutions is obtained for compact potential operators on Hilbert spaces. An application to a second-order boundary value problem is also given as an illustration of our results.

В статье изучаются тождества супералгебр Ли над полем нулевой характеристики. Построена серия примеров конечномерных разрешимых супералгебр с ненильпотентным коммутантом, для которых PI-экспонента роста коразмерностей существует и является целым числом. Библиография: 24 наименования.

This paper is concerned with existence of normalized ground state solutions for the mass supercritical fractional nonlinear Schrödinger equation involving a critical growth in the fractional Sobolev sense. The compactness of Palais-Smale sequences is obtained by a special technique, which borrows from the ideas of Soave (J. Funct. Anal. 279 (6) (2020) 1086102020). This paper represents an extension of previously known results - in the local and the nonlocal cases.

We consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of (2, q)-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

We study the following critical Choquard equation on the Heisenberg group: \begin{equation*} \begin{cases} \displaystyle {-\Delta_H u }={\mu} |u|^{q-2}u+\int_{\Omega} \frac{|u(\eta)|^{Q_{\lambda}^{\ast}}} {|\eta^{-1}\xi|^{\lambda}} d\eta|u|^{Q_{\lambda}^{\ast}-2}u &\mbox{in }\ \Omega, u=0 &\mbox{on }\ \partial\Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{H}^N$ is a smooth bounded domain, $\Delta_H$ is the Kohn-Laplacian on the Heisenberg group $\mathbb{H}^N$, $1"0$, $0<\lambda"

We consider a class of noncooperative Schrödinger-Kirchhoff type system which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration-compactness principle for weighted-variable exponents Sobolev spaces and the principle of symmetric criticality of Krawcewicz and Marzantowicz.

The article is about an elliptic problem defined on a stratified Lie group. Both sub and superlinear cases are considered whose solutions are guaranteed to exist in light of the interplay between the nonlinearities and the weak L 1 datum. The existence of infinitely many solutions is proved for suitable values of λ , p , q by using the Symmetric Mountain Pass Theorem.

We study identities of Lie superalgebras over a field of characteristic zero. We construct a series of examples of finite-dimensional solvable Lie superalgebras with non-nilpotent commutator subalgebra for which the PI-exponent of codimension growth exists and is an integer number.

We consider the following convective Neumann systems: ( S ) -Δ _p_1u_1+ |∇ u_1| ^p_1/u_1+δ _1=f_1(x,u_1,u_2, ∇ u_1,∇ u_2) in Ω , -Δ _p_2u_2+ |∇ u_2| ^p_2/u_2+δ _2=f_2(x,u_1,u_2, ∇ u_1,∇ u_2) in Ω , |∇ u_1| ^p_1-2∂ u_1/∂η=0= |∇ u_2| ^p_2-2∂ u_2/∂η on ∂Ω , where Ω is a bounded domain in ℝ^N ( N≥ 2 ) with a smooth boundary ∂ Ω, δ _1, δ _2 >0 are small parameters, η is the outward unit vector normal to ∂ Ω, f_1, f_2:Ω×ℝ^2×ℝ^2N→ℝ are Carathéodory functions that satisfy certain growth conditions, and Δ _p_i ( 1< p_i< N , i=1,2 ) are the p -Laplace operators Δ _p_iu_i=div(|∇ u_i|^p_i-2∇ u_i) for u_i∈ W^1,p_i(Ω ) . To prove the existence of solutions to such systems, we use a subsupersolution method. We also obtain nodal solutions by constructing appropriate subsolution and supersolution pairs. To the best of our knowledge, such systems have not been studied yet.

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis-Nirenberg problem.

In this article, we deal with the following p p -fractional Schrodinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M([u]ps,A)(-Delta)sp,Au+V(x)|u|p-2u=lambda(integral RN|u|p & lowast;mu,s|x-y|mu dy)|u|p & lowast;mu,s-2u+k|u|q-2u, x is an element of RN where 0<s<1<p, ps<N, p<q<2p & lowast;s,mu, 0<mu<N, lambda, and k are some positive parameters, p & lowast;s,mu=pN-p mu 2N-ps is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions V V and M M satisfy the suitable conditions. By proving the compactness results using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.

In this paper, we study the existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group:M(||u||(p)(mu))(mu(-Delta)(s)(p)u+V(xi)|u|(p-2)u) = f(xi,u)+integral N-H (|u(eta)|Q*)(lambda)/ (|eta-1 xi|)lambda d eta|u|Q* (-2)(lambda )u in H-N, where (Delta)(s)(p) is the fractional p-Laplacian on the Heisenberg group HN, M is the Kirchhoff function, V(xi) is the potential function, 0 < s < 1, 1 < p < (N)/(s) , mu > 0, f (xi,u) is the nonlinear function, 0 < lambda < Q, Q = 2N+2, and Q*(lambda)= (2Q-lambda)/( Q-2) is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if mu is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has m pairs of solutions if mu is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case.

We introduce a novel framework for embedding anisotropic variable exponent Sobolev spaces into spaces of anisotropic variable exponent Hölder-continuous functions within rectangular domains. We establish a foundational approach to extend the concept of Hölder continuity to anisotropic settings with variable exponents, providing deeper insight into the regularity of functions across different directions. Our results not only broaden the understanding of anisotropic function spaces but also open new avenues for applications in mathematical and applied sciences.