Analysis and Mathematical Physics最新文献

Symmetries of the large BKP hierarchy, also known as Toda hierarchy of B type, are investigated in this paper. We firstly construct symmetries of the large BKP hierarchy by the method of additional symmetries. Then we derive Adler–Shiota–van Morebeke formula to link the actions of additional symmetries on Lax operators and tau functions.

In this paper, we obtain new upper bounds for the Lieb–Thirring inequality on the spheres of any dimension greater than 2. As far as we have checked, our results improve previous results found in the literature for all dimensions greater than 2. We also prove and exhibit an explicit new upper bound for the Lieb–Thirring inequality on SO(3). We also discuss these estimates in the case of general compact Lie groups. Originally developed for estimating the sums of moments of negative eigenvalues of the Schrödinger operator in (L^2(mathbb {R}^n)), these inequalities have applications in quantum mechanics and other fields.

Fermat type functional equation with four terms

is difficult to solve completely even if (n=2,3), in which the certain type of the above equation is also interesting and significant. In this paper, we first to consider the Bi-Fermat type quadratic partial differential equation

in (mathbb {C}^{2}). In addition, we consider the Bi-Fermat type cubic difference equation

in (mathbb {C}) and obtain partial meromorphic solutions on the above equation.

In this paper, we study value distribution of meromorphic functions concerning differences and mainly prove the following result: Let f be a transcendental meromorphic function of (1 le rho (f) < infty ), let c be a nonzero constant, n a positive integer, and let P, Q be two polynomials. If (max left{ lambda (f-P), lambda left( frac{1}{f}right) right} <rho (f)) and (Delta _{c}^{n}f not equiv 0), then we have (i) (delta (Q, Delta _c^n f)=0) and (lambda (Delta _{c}^{n}f-Q)=rho (f)), for (Delta _{c}^{n}Pnot equiv Q); (ii) (delta (Q, Delta _c^n f)=1) and (lambda (Delta _{c}^{n}f-Q)<rho (f)), for (Delta _{c}^{n}Pequiv Q). The results obtained in this paper extend and improve some results due to Chen-Shon[J Math Anal Appl 2008], [Sci China Ser A 2009], Liu[Rocky Mountain J Math 2011], Cui-Yang[Acta Math Sci Ser B 2013], Chen[Complex Var Elliptic Equ 2013], Wang-Liu-Fang[Acta Math. Sinica (Chinese Ser) 2016].

We characterize geodesic flows, admitting two commuting quadratic integrals with common principal directions, in terms of the geodesic 4-webs such that the tangents to the web leaves are common zero directions of the integrals. We prove that, under some natural geometric hypothesis, the metric is of Stäckel type.

We firstly describe entire solutions of variation of the well-known PDE of tubular surfaces. In addition, we consider entire solutions of certain partial differential equations, which are related with the Picard’s little theorem. Moreover, we obtain a Tumura-Clunie type theorem in ({mathbb {C}}^{m}), which is an improvement of a result given by Hu-Yang (Bull Aust Math Soc 90: 444-456, 2014).

In this article, we study the double phase elliptic system which contain with the parametric concave-convex nonlinearities and critical growth. The introduction of mixed critical terms brings some difficulties to the problem. For example, in proving that the solution is nontrivial, we need to do an additional series of studies on scalar equation. By introducing a new optimal constant (S_{alpha ,beta }) in the double phase system, considering the different magnitude relationships of the exponential terms, and using the fibering method in form of the Nehari manifold and the Brezis-Lieb Lemma, the existence and multiplicity of solutions in subcritical and critical cases are obtained separately.

This paper investigates the temporal decay rates of solutions to the Cauchy problem of a model, which describes the combustion of the compressible fluid. Suppose that the initial data is a small perturbation near the equilibrium state ((rho _infty , 0,theta _infty ,zeta )), where (rho _infty >0), (theta _infty <theta _I) (the ignition temperature), and (0< zeta leqslant 1), we first establish the global-in-time existence of strong solutions via a standard continuity argument. With the additional (L^1)-integrability of the initial perturbation, we then employ the Fourier theory and the cancellation mechanism of low-medium frequent part to derive the optimal temporal decay rates of all-order derivatives of strong solutions. Our work is a natural continuation of previous result in the case of (theta _infty >theta _I) discussed in Wang and Wen (Sci China Math 65:1199–1228 (2022).

This paper is concerned with the following HLS upper critical focusing Choquard equation with a non-autonomous nonlocal perturbation

where (mu ,c>0), (N ge 3), (0<alpha <N), (2_alpha :=frac{N+alpha }{N}<p<2^*_alpha :=frac{N+alpha }{N-2}), (lambda in mathbb {R}) is a Lagrange multiplier, (I_alpha ) is the Riesz potential and (h:mathbb {R}^Nrightarrow (0,infty )) is a continuous function. Under a class of reasonable assumptions on h, we prove the existence of normalized solutions to the above problem for the case (frac{N+alpha +2}{N}le p<frac{N+alpha }{N-2}) and discuss its asymptotical behaviors as (mu rightarrow 0^+) and (crightarrow 0^+) respectively. When (frac{N+alpha }{N}<p<frac{N+alpha +2}{N}), we obtain the existence of one local minimizer after considering a suitable minimization problem.

The purpose of this article is to prove the existence and uniqueness of weak solutions to the following obstacle problem of p-Laplace-type:

with data belonging to the dual of Sobolev spaces. The main result is demonstrated by means of Kinderlehrer and Stampacchia’s Theorem and Young’s measure theory.