Analysis and Mathematical Physics最新文献

In this paper, we study the existence and multiplicity for the following Schrödinger–Poisson equation

where (a>0) is a prescribed mass, (kappa in {mathbb {R}}setminus {0}) and (lambda in {mathbb {R}}) is an undetermined parameter which appears as a Lagrange multiplier. Our results are threefold: (i) for the case (kappa <0), we obtain the normalized ground state solution for (a>0) small by working on the Pohozaev manifold, where f satisfies the (L^2)-supercritical and Sobolev subcritical conditions, and the behavior of the normalized ground state energy (c_a) is also obtained; (ii) we prove that the above equation possesses infinitely many radial solutions whose energy converges to infinity; (iii) for (kappa >0) and (f(u)=|u|^{4}u), we revisit the Brézis–Nirenberg problem with a nonlocal perturbation and obtain infinitely many radial solutions with negative energy. Our results implement some existing results about the Schrödinger–Poisson equation in the (L^2)-constraint setting.

In this research, we define an essential identity for differentiable functions in the framework of tempered fractional integral. By utilizing this identity, we deduce several modifications of fractional Milne-type inequalities. We provide novel expansions of Milne-type inequalities in the domain of tempered fractional integrals. The investigation emphasises important functional categories, including convex functions, bounded functions, Lipschitzian functions, and functions with bounded variation.

In this article, we establish the existence of ground state solutions for a magnetic critical Choquard Poisson system with multiple potentials by variational methods.

The aim of this paper is first to introduce generalizations of quantum integrals and derivatives which are called ((phi ,-,h)) integrals and ((phi ,-,h)) derivatives, respectively. Then we investigate some implicit integral inequalities for ((phi ,-,h)) integrals. Different classes of convex functions are used to prove these inequalities for symmetric functions. Under certain assumptions, Hermite–Hadamard-type inequalities for q-integrals are deduced. The results presented herein are applicable to convex, m-convex, and (hbar )-convex functions defined on the non-negative part of the real line.

We discuss the zero sets of two-variable polynomials as they relate to an approximation problem in the Hardy space on the bidisk.

A closed form solution for the one-dimensional Schrödinger equation with a finite number of (delta )-interactions

is presented in terms of the solution of the unperturbed equation

and a corresponding transmutation (transformation) operator ({textbf{T}}_{{mathfrak {I}}_{N}}^{f}) is obtained in the form of a Volterra integral operator. With the aid of the spectral parameter power series method, a practical construction of the image of the transmutation operator on a dense set is presented, and it is proved that the operator ({textbf{T}}_{{mathfrak {I}}_{N}}^{f}) transmutes the second derivative into the Schrödinger operator ({textbf{L}}_{q,{mathfrak {I}}_{N}}) on a Sobolev space (H^{2}). A Fourier-Legendre series representation for the integral transmutation kernel is developed, from which a new representation for the solutions and their derivatives, in the form of a Neumann series of Bessel functions, is derived.

The main objective of this paper is to study the p-k-Hessian problems. To our knowledge, the problems that has additional term in the p-k-Hessian operator were seldom studied in the literature. By means of monotone iteration method and Arzelà-Ascoli theorem, this paper investigates the existence of positive radially symmetric solutions of the following augmented p-k-Hessian equations

and p-k-Hessian systems

In this paper, we study the Cauchy problem for the Camassa–Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space (B_{p,1}^{1}cap C^{0,1}) with (pin (2,infty ]) is discontinuous at origin. More precisely, the initial data in (B_{p,1}^{1}cap C^{0,1}) can guarantee that the Camassa–Holm equation has a unique local solution in (W^{1,p}cap C^{0,1}), however, this solution is instable and can have an inflation in (B_{p,1}^{1}cap C^{0,1}).

Let ((X,d,mu )) is a space of homogeneous type, we establish a new class of fractional-type variable weights (A_{p(cdot ), q(cdot )}(X)). Then, we get the new weighted strong-type and weak-type characterizations for fractional maximal operators (M_eta ) on weighted variable Lebesgue spaces over ((X,d,mu )). This study generalizes the results by Cruz-Uribe–Fiorenza–Neugebauer (J Math Anal Appl 64(394):744–760, 2012), Bernardis–Dalmasso–Pradolini (Ann Acad Sci Fenn-M 39:23-50, 2014), Cruz-Uribe–Shukla (Stud Math 242(2):109–139, 2018), and Cruz-Uribe–Cummings (Ann Fenn Math 47(1):457–488, 2022).

The paper is concerned with the p-Kirchhoff equation

where (a,b>0). When (V(x)=0), (p=2) and (Nge 3), we obtain that any energy ground state normalized solutions of (1) has constant sign and is radially symmetric monotone with respect to some point in (mathbb {R}^{N}) by using some energy estimates. When (V(x)not equiv 0, p>sqrt{3}+1, frac{2}{p-2}<ple N<2p), under an explicit smallness assumption on V with (lim _{|x|rightarrow infty }V(x)=sup _{mathbb {R}^{N}}V(x)), we prove the existence of energy ground state normalized solutions of (1).