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)u^p-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?_b∞?_lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<_b∞$b(x)<b_{\infty }$. For any periodic lattice L   in R^N${\mathbb{R}}^N$ and for any b∈C(R^N)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<_b∞$b(x)<b_{\infty }$, _b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination b^Y$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=b^Y(x)u^p-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.