Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity

Abstract

In this paper, we study the existence of least energy solutions to the following fractional Kirchhoff problem with logarithmic nonlinearity M([u]s,pp)(−Δ)psu=h(x)|u|θp−2uln|u|+λ|u|q−2ux∈Ω,u=0x∈RN∖Ω,where s∈(0,1), 1<p<N∕s, Ω⊂RN is a bounded domain with Lipschitz boundary, M([u]s,pp)=[u]s,p(θ−1)p with θ≥1 and [u]s,p is the Gagliardo seminorm of u, h∈C(Ω¯) may change sign, λ>0 is a parameter, q∈(1,ps∗) and (−Δ)ps is the fractional p−Laplacian. When θp<q<ps∗ and h is a positive function on Ω, the existence of least energy solutions is obtained by restricting the discussion on Nehari manifold. When 1<q<θp and h is a sign-changing function on Ω, two local least energy solutions are obtained by using the Nehari manifold approach.