Multispike solutions for a nonlinear elliptic problem involving critical Sobolev exponent

Abstract

The main purpose of this paper is to construct families of positive solutions for the equationon ∂Ω which blow-up and concentrate in k ≥ 1 different points of Ω as ε goes to 0. We exhibit some examples of contractible domains where a large number of solutions exists.

The Ljapunov–Schmidt Reduction for Some Critical Problems

“…Conversely, Rey in [40,41] proved that any C 1 −stable critical point z0 of the Robin's function generates a family of solutions which blows-up at z0 as ǫ goes to zero. MussoPistoia in [35] and Bahri-Li-Rey in [3] studied existence of solutions which blow-up at κ different points of Ω. Grossi-Takahashi [26] proved the nonexistence of positive solutions blowing up at κ ≥ 2 points for these problems in convex domains.…”

“…If the domain has two small holes, Musso-Pistoia in [34] constructed a sign changing solution with one positive blow-up point and one negative blow-up point at the centers of the two holes. Musso-Pistoia in [33] and Ge-Musso-Pistoia in [24] found a large number of sign changing solutions to (C)ǫ: the solutions are a superposition of bubbles with alternating sign whose centers collapse to the center of the hole as ǫ goes to zero.…”

“…In the following we describe the main steps to get some of the previous results. We will refer to [35] and [3,21,31] for the proofs related to the construction of positive and signchanging multi-bubbles to problems (BN )ǫ and (AC)ǫ, respectively. We will refer to [36] and to [33,24] for the proofs related to the construction of towers of bubbles to problems (AC)ǫ and (C)ǫ, respectively.…”

A Neumann problem with critical exponent in nonconvex domains and Lin-Ni’s conjecture

“…In the critical case, for large μ, nonconstant solutions exist [1], [54]. As in the subcritical case the least energy solution blows up, as μ goes to infinity, at a unique point which maximizes the mean curvature of the boundary [3], [42]. Higher energy solutions have also been exhibited, blowing up at one [2], [55], [48], [26] or several separated boundary points [41], [37], [56], The above conjecture was studied by Adimurthi-Yadava [4], [5] and Budd-KnappPeletier [11] in the case Ω = B R (0) and u radial.…”

Concentration phenomena for a fourth-order equation on ℝn

“…In the past few decades, there has been a wide range of activity in the study of concentration phenomena for second-order elliptic equations involving critical Sobolev exponent; see for instance [Atkinson and Peletier 1987;Bahri et al 1995;Ben Ayed et al 2003;Brezis and Peletier 1989;Chabrowski and Yan 1999;del Pino et al 2002;2003;Han 1991;Micheletti and Pistoia 2003;Musso and Pistoia 2002;Rey 1989;1990;1992;1999] and the references therein. In sharp contrast to this, very little is known for equations involving the biharmonic operator.…”