Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

We consider the elliptic problem −Δu+ u= b(x)|u|p−2u+ h(x) in Ω, u∈H1 0 (Ω), where 2 < p < (2N/(N − 2)) (N ≥ 3), 2 < p <∞ (N = 2), Ω is a smooth unbounded domain in RN , b(x) ∈ C(Ω), and h(x) ∈ H−1(Ω). We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b(x) satisfies b(x)→ b∞ > 0 as |x| →∞ and b(x)≥ c for some suitable constants c ∈ (0,b∞), and h(x)≡ 0. Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b(x) also satisfies the above conditions, h(x)≥ 0 and 0 < ‖h‖H−1 < (p− 2)(1/(p− 1))[bsupSp(Ω)], where S(Ω) is the best Sobolev constant of subcritical operator in H 0 (Ω) and bsup = supx∈Ωb(x).