New Existence of Multiple Solutions for Fractional Kirchhoff Equations with Logarithmic Nonlinearity

By using the Ekeland variational principle and Nehari manifold, we study the following fractional p-Laplacian Kirchhoff equations: M[u]s,pp+∫RNV(x)|u|pdx[(−Δ)psu+V(x)|u|p−2u]=λ|u|q−2uln|u|,x∈RN,(P). In these equations, λ∈R∖{0},p∈(1,+∞), s∈(0,1),sp<N,ps*=NpN−sp, M(τ)=a+bτθ−1, a,b∈R+,1<θ<ps*p, V(x)∈C(RN,R) is a potential function and (−Δ)ps is the fractional p-Laplacian operator. The existence of solutions is deeply influenced by the positive and negative signs of λ. More precisely, (i) Equation (P) has one ground state solution for λ>0 and pθ<q<ps*, with a positive corresponding energy value; and (ii) Equation (P) has at least two nontrivial solutions for λ<0 and p<q<ps*, with positive and negative corresponding energy values, respectively.