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. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)up-1-Δu+u=b(x)up-1, u>0u>0, u∈H1(RN)u∈H1(RN), p∈(2,2N/(N-2))p∈(2,2N/(N-2)) was proved under assumption b(x)⩾b∞≔lim|x|→∞b(x)b(x)⩾b∞≔lim|x|→∞b(x). In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)0b∞>0, there is a finite set Y⊂LY⊂L and a convex combination bYbY of b(·-y)b(·-y), y∈Yy∈Y, such that the problem -Δu+u=bY(x)up-1-Δu+u=bY(x)up-1 has a positive solution u∈H1(RN)u∈H1(RN).