INFINITELY MANY SOLUTIONS FOR NONLOCAL SYSTEMS INVOLVING FRACTIONAL LAPLACIAN UNDER NONCOMPACT SETTINGS

In this paper, we study a class of Brezis–Nirenberg problems for nonlocal systems, involving the fractional Laplacian $(-\unicode[STIX]{x1D6E5})^{s}$ operator, for $0<s<1$ , posed on settings in which Sobolev trace embedding is noncompact. We prove the existence of infinitely many solutions in large dimension, namely when $N>6s$ , by employing critical point theory and concentration estimates.

A bifurcation result involving Sobolev trace embedding and the duality mapping of W1,p

We consider the perturbed nonlinear boundary condition problem$$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$Using the Sobolev trace embedding and the duality mapping defined on W1,p(Ω), we prove that this problem bifurcates from the principal eigenvalue λ1 of the eigenvalue problem$$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$

THE BEST SOBOLEV TRACE CONSTANT IN PERIODIC MEDIA FOR CRITICAL AND SUBCRITICAL EXPONENTS

In this paper we study homogenisation problems for Sobolev trace embedding H1(Ω) ↪ Lq(∂Ω) in a bounded smooth domain. When q = 2 this leads to a Steklov-like eigenvalue problem. We deal with the best constant of the Sobolev trace embedding in rapidly oscillating periodic media, and we consider H1 and Lq spaces with weights that are periodic in space. We find that extremals for these embeddings converge to a solution of a homogenised limit problem, and the best trace constant converges to a homogenised best trace constant. Our results are in fact more general; we can also consider general operators of the form aɛ(x, ∇u) with non-linear Neumann boundary conditions. In particular, we can deal with the embedding W1,p(Ω) ↪ Lq(∂Ω).

AN OPTIMIZATION PROBLEM RELATED TO THE BEST SOBOLEV TRACE CONSTANT IN THIN DOMAINS

Let Ω ⊂ ℝNbe a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) ↪ Lq(∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem [Formula: see text] for functions that verify u|A= 0. It is known that there exists an optimal hole that minimizes the best constant SAamong subsets of Ω of the prescribed volume.In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain.

First Variations of the Best Sobolev Trace Constant with Respect to the Domain

In this paper we study the best constant of the Sobolev trace embedding H1(Ω) → L2(∂Ω), where Ω is a bounded smooth domain in ℝN. We find a formula for the first variation of the best constant with respect to the domain. As a consequence, we prove that the ball is a critical domain when we consider deformations that preserve volume.

The best Sobolev trace constant in domains with holes for critical or subcritical exponents

In this paper we study the best constant in the Sobolev trace embedding H1 (Ω) →Lq(∂Ω) in a bounded smooth domain for 1 < q < 2+ = 2(N - 1)/(N - 2), that is, critical or subcritical q. First, we consider a domain with periodically distributed holes inside which we impose that the involved functions vanish. There exists a critical size of the holes for which the limit problem has an extra term. For sizes larger than critical the best trace constant diverges to infinity and for sizes smaller than critical it converges to the best constant in the domain without holes. Also, we study the problem with the holes located on the boundary of the domain. In this case another critical exists and its extra term appears on the boundary.