A Brezis-Nirenberg type theorem on local minimizers for p(x)-Kirchhoff Dirichlet problems and applications

A Note on Borderline Brezis-nirenberg Type Problems

where LAu = div(A(x)∇u) and La,pu = div(a(x)|∇u| ∇u) are, respectively, linear and quasilinear uniformly elliptic operators in divergence form in a non-smooth bounded open subset Ω of R, 1 < p < n, p∗ = np/(n − p) is the critical Sobolev exponent and λ is a real parameter. Both problems have been quite studied when the ellipticity of LA and La,p concentrate in the interior of Ω. We here focus o...

The Brezis-nirenberg Problem for Nonlocal Systems

By means of variational methods we investigate existence, non-existence as well as regularity of weak solutions for a system of nonlocal equations involving the fractional laplacian operator and with nonlinearity reaching the critical growth and interacting, in a suitable sense, with the spectrum of the operator.

THE BREZIS-NIRENBERG PROBLEM FOR THE FRACTIONAL p-LAPLACIAN

We obtain nontrivial solutions to the Brezis-Nirenberg problem for the fractional p-Laplacian operator, extending some results in the literature for the fractional Laplacian. The quasilinear case presents two serious new difficulties. First an explicit formula for a minimizer in the fractional Sobolev inequality is not available when p 6= 2. We get around this difficulty by working with certain...

ON THE LOCAL NIRENBERG PROBLEM FOR σk-TYPE CURVATURES

Maximizers for Gagliardo–nirenberg Inequalities and Related Non-local Problems

In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo–Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb’s Translation Lemma and a Riesz energy version of the Brézis–Lieb lemma.