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 20 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).