A note on a result of M. Grossi

Abstract

Abstract. The purpose of this note is to present a fact complementary to a result in a recent paper of M. Grossi. Making use of an energy balance identity, it is shown that the sufficient conditions for existence of solutions proved in Grossi's paper are also almost necessary.

Multi-layer radial solutions for a supercritical Neumann problem

“…Proof. The existence of ξ, ζ satisfying (2.6) and (2.7) is proved in [13, Lemma 6.1], based on the case of Dirichlet boundary conditions which can be found in [8,Appendix]. Following step by step the last mentioned paper, one can also check that ξ is bounded and that (2.8)-(2.10) hold.…”

Positive constrained minimizers for supercritical problems in the ball

“…(6.3)Catrina provides, via an approximation method, a positive function ϕ(s) which satisfies ϕ(s) (6.3). The function ξ(r) = ϕ(r 2−n )/(n − 2) solves (6.1) and moreover, as shown in[8], lim r→0 + ξ ′ (r) = 0. Next we set ψ(s) = ϕ(s) the assumption V ≥ 0, V ≡ 0.…”

“…(6The same result holds in case of Dirichlet boundary conditions at r = 1, that is ξ ′ (0) = ζ(1) = 0.Proof. In case of Dirichlet boundary conditions the result is proved by Catrina in[8], Appendix. Let us adapt the proof to the case of Neumann boundary conditions.…”

“…Actually, our technique gives a unified proof for both Dirichlet and Neumann boundary conditions. Moreover, again in the Dirichlet case, Catrina proved in [8] that the condition in Theorem 1.4 is "almost" necessary. Indeed he proved the following result.…”

“…We are interested when the exponent p is large. Recent results (see [14,8]) suggest that the existence of solutions of (1.1) is related to the critical points of a function F (r), associated to this equation in the limit as p → +∞ (see (1.3) below). Our aim is to extend the known existence results in this direction through a better understanding of F (r).…”