Gilberto de Assis Pereira

Asymptotics for the best Sobolev constants and their extremal functions

Let p > 1 and let be a bounded and smooth domain of R N , N ≥ 2. It is well known that the infimum λq() := inf n k∇uk p : u ∈ W 1,p 0 () and kuk q = 1 o 0 () and also in C()). Moreover, we prove that any minimizer up ofp() satisfies −�pup = up(xp)�p()δxp , where δxp is the Dirac delta distribution concentrated at the only point xp satisfying |up(xp)| = kupk ∞ = 1. In the second part of the paper we prove that limp→∞ �p() 1 p = 1 kρk1 where ρ denotes the distance function to the boundary ∂. We also prove that there exist pn → ∞, x∗ ∈ and u∞ ∈ W 1,∞ 0 () such that: ρ(x∗) = kρk ∞ , xpn → x∗, upn → u∞ uniformly in , 0 < u∞ ≤ ρ kρk1 in and

Fractional Sobolev inequalities associated with singular problems

In this paper we study Sobolev‐type inequalities associated with singular problems for the fractional p‐Laplacian operator in a bounded domain of RN, N≥2.

On a singular minimizing problem

AbstractFor each q ∈ (0, 1), let

Remarks about a fractional Choquard equation: Ground state, regularity and polynomial decay

{\lambda _q}(\Omega ): = inf\{ ||{\nabla _v}||_{{L^P}(\Omega )}^P:v \in W_0^{1,P}(\Omega )and\int_\Omega {|v{|^q}dx = 1\} }

Remarks about a generalized pseudo-relativistic Hartree equation

λq(Ω):=inf{||∇v||LP(Ω)P:v∈W01,P(Ω)and∫Ω|v|qdx=1} where p > 1 and Ω is a bounded and smooth domain of RN, N ≥ 2. We first show that

Ground state of a magnetic nonlinear Choquard equation

0 < \mu (\Omega ): = \mathop {\lim }\limits_{q \to {0^ + }} {\lambda _q}(\Omega )|\Omega {|^{p/q}} < \infty