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

Abstract

Abstract. We consider the following nonlinear Neumann problem:where Ω ⊂ R N is a smooth and bounded domain, μ > 0 and n denotes the outward unit normal vector of ∂Ω. Lin and Ni (1986) conjectured that for μ small, all solutions are constants. We show that this conjecture is false for all dimensions in some (partially symmetric) nonconvex domains Ω. Furthermore, we prove that for any fixed μ, there are infinitely many positive solutions, whose energy can be made arbitrarily large. This seems to be a new phenomenon for elliptic problems in bounded domains.

Infinitely many sign-changing solutions for a class of critical elliptic systems with Neumann conditions

“…When Ω is a special non-convex domain, Wang et al . in [43] refuted the conjecture of Li and Ni. On the other hand, this conjecture was proved for some different situations (see, for example, [3,5,32]).…”

Infinitely many positive solutions for a Schrödinger–Poisson system

“…This method has also been applied for the study of different problems (see for example [18,19]). In our case, however, many technical difficulties arise due to the presence of the non-local term φ u and a more careful analysis of the interaction between the bumps is required.…”

Infinitely many solutions for the Hénon equation with critical exponent in non-convex domains

“…According to Lemmas A.3 and A.4 in [19] we have known that According to Lemmas A.3 and A.4 in [19] we have known that…”

“…This idea is motivated by the recent paper [19], where infinitely many solutions to a nonlinear elliptic Neumann problem were constructed. In all the singularly perturbed problems, some small parameters are present either in the operator or in the nonlinearity or in the boundary condition.…”

“…The proof can be found in [16,19], Appendix A, and here we do not repeat it again. As a direct consequence of Lemma 3.3, we have the following lemma.…”