Analysis and Mathematical Physics最新文献

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).

Let (kge 1), (0le alpha <d) and (mathfrak {M}_{b,alpha }^k) be the k-th order fractional maximal commutator. When (alpha =0), we denote (mathfrak {M}_{b,alpha }^k=mathfrak {M}_{b}^k). We show that (mathfrak {M}_{b,alpha }^k) is bounded from the first order Sobolev space (W^{1,p_1}(mathbb {R}^d)) to (W^{1,p}(mathbb {R}^d)) where (1<p_1,p_2,p<infty ), (1/p=1/p_1+k/p_2-alpha /d). We also prove that if (0<s<1), (1<p_1,p_2,p,q<infty ) and (1/p=1/p_1+k/p_2), then (mathfrak {M}_b^k) is bounded and continuous from the fractional Sobolev space (W^{s,p_1}(mathbb {R}^d)) to ({W^{s,p}(mathbb {R}^d)}) if (bin W^{s,p_2}(mathbb {R}^d)), from the inhomogeneous Triebel–Lizorkin space (F_s^{p_1,q}(mathbb {R}^d)) to (F_s^{p,q}(mathbb {R}^d)) if (bin F_s^{p_2,q} (mathbb {R}^d)) and from the inhomogeneous Besov space (B_s^{p_1,q}(mathbb {R}^d)) to (B_s^{p,q}(mathbb {R}^d)) if (bin B_s^{p_2,q}(mathbb {R}^d)). It should be pointed out that the main ingredients of proving the above results are some refined and complex difference estimates of higher order maximal commutators as well as some characterizations of the Sobolev spaces, Triebel–Lizorkin spaces and Besov spaces.

In this paper, we focus on investigating the entire solutions to one certain type of non-linear binomial differential equations with respect to several problems posed by Gundersen and Yang. We also illustrate the exponential polynomials solutions to this equation. Some examples are used to illustrate our results.