We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and the electrorheological fluids. also get singular limit formula extending Nguyen results to anisotropic case.

The present paper deals with a Kirchhoff problem under homogeneous Dirichlet boundary conditions, set in a bounded smooth domain Ω of R^{N}. The problem studied is a stationary version of the orig inal Kirchhoff equation, involving the p(x)-Lap lacian operator, in the framework of the variable exponent Lebesgue and Sobolev spaces. The question of the existence of weak solutions is treated. Appl...

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent λ = n−α (that is for the case of α > n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequ...

<p style='text-indent:20px;'>In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence radial ground state solutions for by using refined Sobolev inequality Morrey norm, show that any nonnegative weak have additional decay properties in terms representation.</p>

We study quasi–Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of ε depends on ε−1 and the dimension s. Strong tractability means that it does not depend on s and is bounded by a polynomial in ε−1. The least po...

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation $ \begin{equation*} u_{t} = \Delta u + |u|^{p} \end{equation*} $ on complete Riemannian manifolds dimension n \geq 5 with nonnegative Ricci tensor, when p is smaller than critical Sobolev exponent \frac{n+2}{n-2} $.

This paper is devoted to study a fractional Choquard problem with slightly subcritical exponents on bounded domains. When the exponent of convolution type nonlinearity tends critical one in sense Hardy–Littlewood–Sobolev inequality, we obtain existence multiple positive solutions via Lusternik–Schnirelmann category and nonlocal global compactness. Moreover, prove that topology domain furnishes ...

where a ( x ) is a given function on M . The original interest in such questions grew out of Yamabe's problem (see [40], [39], [2], [27], [15]) which corresponds to the special case where a ( x ) = ( ( N 2)/4(N l ) ) R ( x ) and R ( x ) is the scalar curvature of M . It turns out that, despite its simple form, equation (1) (or ( 2 ) ) has a very rich structure and provides an amazing source of ...