We prove the multiplicity and concentration of normalized solutions of critical biharmonic equations with combined nonlinearities in \mathbb{R}^{N} , \begin{equation*}\Delta^{2}u+V(\varepsilon x)u=\lambda u+\mu |u|^{q-2}u+|u|^{2^{**}-2}u \text{ in }\mathbb{R}^{N}, \quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c^{2},\end{equation*} where \Delta^{2} is the biharmonic operator, N\geq5 , \mu,c>0 , \varepsilon>0 , \lambda\in\mathbb{R} , q\in(2,2+\frac{8}{N}) , and 2^{**}=\frac{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 \varepsilon 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 \mathbb{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.

Let G be a locally compact group, and let (\Phi,\Psi) be a complementary pair of Young functions satisfying the \Delta_{2} -condition. In this article, we consider PM_{\Psi}(G) the Banach algebra of \Psi -pseudomeasures along with its predual, the Orlicz Figà-Talamanca Herz algebra A_{\Phi}(G) . We prove sufficient conditions for the amenability of G in terms of the norm closed topologically invariant subspaces of PM_{\Psi}(G) . Further, when G is amenable and the Young function \Phi satisfies the MA condition, we establish a one-to-one correspondence between certain topologically invariant subalgebras of PM_{\Psi}(G) and the closed subgroups of G . A similar result is obtained for A_{\Phi}(G) , where we derive a bijection between certain topologically invariant subalgebras of A_{\Phi}(G) and the compact subgroups of G .

In this paper, we establish the equivalence of weak and viscosity solutions for a homogeneous problem involving a mixed local and nonlocal elliptic operator in a bounded domain \Omega\subset\mathbb{R}^{N} with Lipschitz boundary. We employ a comparison principle and a priori variational estimates to prove that continuous weak solutions are viscosity solutions and bounded viscosity solutions that vanish outside \Omega are weak solutions. Our results are novel and new for mixed local and nonlocal operators, even for p=2 .

We consider a Dirichlet problem driven by a differential operator with unbalanced growth and a reaction exhibiting the combined effects of a parametric singular term and a resonant perturbation. Using a combination of variational tools and critical groups, we show that, for all small values of the parameter, the problem has at least two bounded positive solutions.

Let B be a Banach space and X a quasi-Banach function lattice. In this paper, we introduce the B -valued weak martingale Hardy spaces associated with X . We then establish the \infty -atomic characterizations of these B -valued weak martingale Hardy-type spaces. As an application, we obtain the boundedness of the \sigma -sublinear operators from B -valued weak martingale Hardy type spaces to weak quasi-Banach function spaces. Using this, we further establish the relationships between different Banach-valued weak martingale Hardy spaces.

This paper mainly aims to demonstrate the boundedness for commutators of fractional maximal operator and sharp maximal operator in the slice spaces, where the symbols of the commutators belong to the \mathrm{BMO} (bounded mean oscillation) spaces. Meanwhile, some new characterizations for \mathrm{BMO} spaces are given.

We show that every graph continuous family of unbounded operators in a Hilbert space becomes Riesz continuous after one-sided multiplication by an appropriate family of unitary operators. This result provides a simple definition of the index for graph continuous families of Fredholm operators, and we show that for such families this index coincides with the index defined by N. Ivanov (2021). This result also has two corollaries for operators with compact resolvents: (1) the identity map between the space of such operators with the Riesz topology and the space of such operators with the graph topology is a homotopy equivalence; (2) every graph continuous family of such operators acting between fibers of Hilbert bundles becomes Riesz continuous in appropriate trivializations of the bundles.For self-adjoint operators, multiplication by unitary operators should be replaced by conjugation. In general, a graph continuous family of self-adjoint operators with compact resolvents cannot be made Riesz continuous by an appropriate conjugation. We obtain a partial analog of the “trivialization” result above for self-adjoint operators and describe obstructions to the existence of such a trivialization in the general case. This motivates the notion of a polarization of a Hilbert bundle, and we prove a similar result for polarizations. These results are closely related to the recent work of Ivanov (2021) and provide alternative proofs for some of his results. We then show that, under a minor assumption on the space of parameters and for operators which are neither essentially positive nor essentially negative, there is always a trivialization making the family Riesz continuous.

This paper deals with a 4th-order quasilinear hyperbolic equation involving strong damping and superlinear source, u_{tt}-\Delta_{m}u+\Delta^{2}u-\Delta_{r}u_{t}=|u|^{p-2}u, \quad (x,t)\in \Omega\times(0,T_{\max}), subject to homogeneous Navier boundary condition, where \Omega is an open bounded domain in \mathbb{R}^{n} (n>2) ; p>m\geq r\geq 2 ; \Delta_{m}u:=\mathrm{div}(|\nabla u|^{m-2}\nabla u) ; and \Delta_{r}u_{t}:=\mathrm{div}(|\nabla u_{t}|^{r-2}\nabla u_{t}) . For the positive initial energy case, we obtain the existence of global solutions, where the decay estimates are divided into five kinds for all the exponent regions. When the initial energy is negative, we arrive at the upper and lower bounds of blow-up time. The L^{2} inner product (u_{1},u_{0})>0 of the initial data is not a necessary condition on the existence of blow-up solutions in the region \{p>m>2=r\} .

Our aim in this paper is to establish integrabilities for generalized Riesz potentials R_{\alpha,m;x_0}f(x) = \int_{\R^n}R_{\alpha,m;x_0}(x,y) f(y) dy of functions f in weighted Morrey–Orlicz spaces, where x_{0}\in \R^{n} , R_{\alpha}(x) = |x|^{\alpha -n} , and R_{\alpha,m;x_0}(x,y) = R_{\alpha}(x-y) - \sum_{|\ell| \le m-1}\frac{(x-x_{0})^{\ell}}{\ell !}(D^{\ell}R_{\alpha})(x_{0}-y), which was first introduced by Hayman for the integral representation of subharmonic functions and then by the first author for the integral representation of Sobolev functions. As an application, we discuss integral mean differentiability of Riesz potentials with exceptional sets whose size are evaluated by (\alpha,p) -capacity.

In this paper, we investigate the reverse isoperimetric inequality with respect to the Gaussian measure for convex sets in \mathbb{R}^{2} . While the isoperimetric problem for the Gaussian measure is well understood, many relevant aspects of the reverse problem have not yet been investigated. In particular, to the best of our knowledge, there seem to be no results on the shape that the isoperimetric set should take. Here, through a local perturbation analysis, we show that smooth perimeter-maximizing sets have locally flat boundaries. Additionally, we derive sharper perimeter bounds than those previously known, particularly for specific classes of convex sets, such as the convex sets symmetric with respect to the axes. Finally, for quadrilaterals with vertices on the coordinate axes, we prove that the set maximizing the perimeter “degenerates” into the x -axis, traversed twice.