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In this paper, we study the L p -boundedness of the commutator [ b , S R δ (H) ] (f) = b S R δ (H) f - S R δ (H) (b f) of a BMO function b and the Bochner–Riesz means S R δ (H) for Hermite operator H = - Δ + | x | 2 on R n , n ≥ 2 . We show that if δ > δ (q) = n (1 / q - 1 / 2) - 1 / 2 , the commutator [ b , S R δ (H) ] is bounded on L p (R n) whenever q < p < q ′ and 1 ≤ q ≤ 2 n / (n + 2) . [ABSTRACT FROM AUTHOR]

We are concerned with the following class of equations with exponential nonlinearities: which is related to the Tzitzéica equation. Here $$h_1, h_2$$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result. [ABSTRACT FROM AUTHOR]

A new technique is proposed for the solution of the Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient $${G(t) = \alpha_{1}(t)I + \alpha_{2}(t)Q(t)}$$ , $${\alpha_{1}(t), \alpha_{2}(t) \in H(L)}$$ , I = diag{1, 1}, Q( t) is a $${2\times2}$$ zero-trace polynomial matrix. This problem has numerous applications in elasticity and diffraction theory. The main feature of the method is the removal of essential singularities of the solution to the associated homogeneous scalar Riemann-Hilbert problem on the hyperelliptic surface of an algebraic function by means of the Baker-Akhiezer function. The consequent application of this function for the derivation of the general solution to the vector Riemann-Hilbert problem requires the finding of the $${\rho}$$ zeros of the Baker-Akhiezer function ( $${\rho}$$ is the genus of the surface). These zeros are recovered through the solution to the associated Jacobi problem of inversion of abelian integrals or, equivalently, the determination of the zeros of the associated degree- $${\rho}$$ polynomial and solution of a certain linear algebraic system of $${\rho}$$ equations. [ABSTRACT FROM AUTHOR]

This work focuses on the second type of generalized Feigenbaum's equation where φ( x) is C-increasing function on [0, 1] and satisfies that φ(0) = 0,0 < φ′( x) < 1 ( x ∈ [0, 1]). Using constructive method, we discuss the existence of C-single-valley solutions whose derivatives are not equal to 0 on origin of the above equation. [ABSTRACT FROM AUTHOR]