A semilinear elliptic problem on unbounded domains with reverse penalty

✍ Scribed by Kyril Tintarev

• Publisher: Elsevier Science
• Year: 2006
• Tongue: English
• Weight: 119 KB
• Volume: 64
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2005.07.003

✦ Synopsis

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

was proved under assumption b(x) b ∞ := lim |x|→∞ b(x). In this paper we prove the existence for certain functions b satisfying the reverse inequality b(x) < b ∞ . For any periodic lattice L in R N and for any b ∈ C(R N ) satisfying b(x) < b ∞ , b ∞ > 0, there is a finite set Y ⊂ L and a convex combination b Y of b(•-y), y ∈ Y , such that the problem -u+u=b Y (x)u p-1 has a positive solution u ∈ H 1 (R N ).