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

Abstract

We consider the elliptic problem −Δu+u=b(x)|u|p−2u+h(x) in Ω, u∈H01(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))(p−1)/(p−2)[bsupSp(Ω)]1/(2−p), where S(Ω) is the best Sobolev constant of subcritical operator in H01(Ω) and bsup=supx∈Ωb(x).