On indefinite Kirchhoff-type equations under the combined effect of linear and superlinear terms

Abstract

We investigate the eigenvalue problem for Kirchhoff type equations involving a superlinear perturbation, namely, −a∫RN|∇u|2dx+1Δu+μV(x)u=λf(x)u+g(x)|u|p−2u in RN, where V∈C(RN) is a potential well with the bottom Ω≔int{x∈RN|V(x)=0}. When N = 3 and 4 < p < 6, for each a > 0 and μ sufficiently large, we obtain at least one positive solution for 0 < λ ≤ λ1(fΩ), while at least two positive solutions exist for λ1(fΩ) < λ < λ1(fΩ) + δa without any assumption on the integral ∫Ωg(x)ϕ1pdx, where λ1(fΩ) > 0 is the principal eigenvalue of −Δ in H01(Ω) with the weight function fΩ ≔ f|Ω and ϕ1 > 0 is the corresponding principal eigenfunction. When N ≥ 3 and 2 < p < min{4, 2*}, for μ sufficiently large, we conclude that (i) at least two positive solutions exist for 0 < a < a0(p) and 0 < λ < λ1(fΩ); (ii) if ∫Ωg(x)ϕ1pdx<0, at least three positive solutions exist for 0 < a < a0(p) and λ1(fΩ)≤λ<λ1(fΩ)+δ̄a; and (iii) if ∫Ωg(x)ϕ1pdx>0, at least two positive solutions exist for a ≥ a0(p) and 0≤λa+<λ<λ1(fΩ).