We prove the L estimate for isotropic version of homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region smooth potentials range under values interaction exponent (2), weighted Poincaré inequality is natural consequence traditional Hardy inequality, turn implies that norms solutions propagate L1 space. Now, based on work De Giorgi, Nash, Moser, as well...

A finite-dimensional global attractor A can be embedded, using some linear map L, into a Euclidean space R k of sufficiently high dimension. The Holder exponent ̈ y1 Ž . of L depends upon k and upon t A , the ‘‘thickness exponent’’ of A. We show that global attractors which are uniformly bounded in the Sobolev spaces H s for Ž . all s ) 0 have t A s 0. It follows, using a result of B. R. Hunt an...

We consider the following nonlinear singular elliptic equation −div (|x| −2a ∇u) = K(x)|x| −bp |u| p−2 u + λg(x) in R N , where g belongs to an appropriate weighted Sobolev space, and p denotes the Caffarelli–Kohn– Nirenberg critical exponent associated to a, b, and N. Under some natural assumptions on the positive potential K(x) we establish the existence of some λ 0 > 0 such that the above pr...

see for instance [29]. Notice that the exponent p or q could be larger than N+2 N−2 . Hence the usual Sobolev space H 0 (Ω)×H1 0 (Ω) is not suitable to handle the problem. To study the problem (1.2) under the condition (1.3), a key observation was done by Hulshof and Van de Vorst [16], De Figueiredo and Felmer [9]. In order to solve this problem, the main idea is to destroy the symmetry between...

where ua = 1 vol(Mn) ∫ Mn u, p∗ = np/(n − p) is the Sobolev conjugate of p. This inequality can be proved by combining Sobolev inequality with Poincaré inequality, see, for example, Hebey’s book [8]. In this paper we are interested in the estimates of the best constant and the existence of extremal functions to the above inequality. Analytically these are naturally motivated questions. On the o...

We derive some of the basic properties of weighted variable exponent Lebesgue spaces L p(.) w (R) and investigate embeddings of these spaces under some conditions. Also a new family of Wiener amalgam spaces W (L p(.) w , L q υ) is defined, where the local component is a weighted variable exponent Lebesgue space L p(.) w (R) and the global component is a weighted Lebesgue space Lυ (R) . We inves...