Bubbling along boundary geodesics near the second critical exponent

Abstract

Abstract. The role of the second critical exponent p = (n + 1)/(n − 3), the Sobolev critical exponent in one dimension less, is investigated for the classical Lane-Emden-Fowler problem u+ u p = 0, u > 0 under zero Dirichlet boundary conditions, in a domain in R n with bounded, smooth boundary. Given , a geodesic of the boundary with negative inner normal curvature we find that for p = (n + 1)/(n − 3) − ε, there exists a solution u ε such that |∇u ε | 2 converges weakly to a Dirac measure on as ε → 0 + , provided that is nondegenerate in the sense of second variations of length and ε remains away from a certain explicit discrete set of values for which a resonance phenomenon takes place.

Curve-Like Concentration Layers for a Singularly Perturbed Nonlinear Problem with Critical Exponents

“…This leads to a complicated resonance for the construction of the solution. The proof of our result is based on a sort of infinite Liapunov Schmidt reduction method, used in other contexts like [10,13], which is close in spirit to that of finite dimensional Liapunov Schmidt reduction. This method helps us deal with the complicated resonance, which also appears in the construction of concentration for the Schrödinger equation in [10].…”

“…Let us mention that to our knowledge no results for solutions to (1.9) concentrating along a high dimensional set on the boundary is known so far. For a related nonlinear Downloaded by [University of Southern Queensland] at 02: 22 15 March 2015 boundary value problem, with Dirichlet boundary condition, at the second critical exponent, we refer the reader to [13].…”

“…Now we consider the linearization of the problem (3.1) at W x for = 1, say W 0 . It is proved in [13] that there exists a unique positive eigenvalue 0 with corresponding eigenfunction Z 0 (even) in L 2 N of the problem…”

“…In this section, we use a gluing technique (as in [12,13], to reduce the problem (2.2) in to a projected nonlinear problem on the infinite strip defined in (3.23) with the coordinates s z defined in (2.5).…”

“…In this section we will use the weighted space in [13] to develop the linear resolution theory and also the method in [11] to deal with the boundary error.…”