In this brief note, we study the problem of asymptotic behavior solutions for non-resonant, singularly perturbed linear Neumann boundary value problems εy″+ky=f(t), y′(a)=0, y′(b)=0, k>0, with an indication possible extension to more complex cases. Our approach is based on analysis integral equation associated problem.

Au — 0 in D; u = ƒ on bD9 where ƒ and its gradient on 3D belong to L(do). For C domains, these estimates were obtained by A. P. Calderón et al. [1]. For dimension 2, see (d) below. In [4] and [5] we found an elementary integral formula (7) and used it to prove a theorem of Dahlberg (Theorem 1) on Lipschitz domains. Unknown to us, this formula had already been discovered long ago by Payne and We...

Here, and throughout the paper, we useD to denote the α-order derivative, where α is a multi-index and we are using standard multi-index notation. Moreover, γα := |α|!/α! denotes the polynomial coefficient. [The naturality of this choice of the Sobolev inner product will be pointed out and discussed below.] We define the Sobolev space W (Ω) to be the closure of C(Ω̄) with respect to the above in...

In this paper we consider a Neumann problem of the type (Pλ) 8< : −∆u = α(x)(|u|q−2u− u) + λf(x, u) in Ω, ∂u ∂ν = 0 on ∂Ω. Applying the theory developed in [13], we establish, under suitable assumptions, the existence of an open interval Λ ⊆ R and of a positive real number %, such that, for each λ ∈ Λ, problem (Pλ) admits at least three weak solutions in W 1,2(Ω) whose norms are less than %. Le...