A semilinear elliptic problem on unbounded domains with reverse penalty

In the well-known work of P.-L. Lions [The concentration-compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincare, Analyse Non Lineaire 1 (1984) 109-1453] existence of positive solutions to the equation -Delta u + u =b(x)u(p-1), u > 0, u is an element of H-1 (R-N), p is an element of (2, 2N/(N - 2)) was proved under assumption b(x) >= b(infinity) := lim(vertical bar x vertical bar ->infinity)b(x). In this paper we prove the existence for certain functions b satisfying the reverse inequality b(x) < b(infinity). For any periodic lattice L in R-N and for any b c C(RN) satisfying b(x) < b(infinity), b(infinity) > 0, there is a finite set Y subset of L and a convex combination b(Y) of b(.-y), y is an element of Y, such that the problem -Delta u +u = b(Y) (x)u(p-1) has a positive solution u is an element of H-1 (R-N). (c) 2005 Elsevier Ltd. All rights reserved.