and Applied Analysis 3 Theorem 1.1. Assume that (A1) and (B1) hold. If λ ∈ 0,Λ0 , then Eλa,b admits at least one positive solution inH1 R . Associated with Eλa,b , we consider the energy functional Jλa,b inH1 R : Jλa,b u 1 2 ‖u‖H1 − λ q ∫ RN a x |u|dx − 1 p ∫

In this paper, we prove an existence theorem for the pseudo-nonlocal Cauchy problem x′(t) + Ax(t) = f(t, x(t), ∫ t t0 k(t, s, x(s))ds), x0(t0) = x0−g(x), where A is the infinitesimal generator of a C0 semigroup of operator {T (t)}t>0 on a Banach space. The functions f, g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.

Extending recent work for the linear Poisson problem for the Laplacian in the framework of Sobolev-Besov spaces on Lipschitz domains by Jerison and Kenig [16], Fabes, Mendez and Mitrea [9], and Mitrea and Taylor [30], here we take up the task of developing a similar sharp theory for semilinear problems of the type ∆u − N(x, u) = F (x), equipped with Dirichlet and Neumann boundary conditions.

In this paper we study the global (in time) existence of small data Sobolev solutions and blow-up to Cauchy problem for semilinear evolution models with scale-invariant friction, visco-elastic damping power nonlinearity. We are interested in critical exponents question how higher regularity influences admissible range $ p nonlinearity get solutions.