Solvability Of A P(x)-Kirchhoff Equation With Concave And Convex Nonlinear Term

In this paper,the solvability of a class of p(x)-Kirchhoff equations with concaveconvex nonlinearities are studied by using perturbation method,mountain pass lemma,Ekeland’s variational principle,a priori estimate and Nehari manifold.On the one hand,we consider the following p(x)-Kirchhoff equation with concaveconvex nonlinearities where 0 ∈Ω(?)RN is a bounded domain with smooth boundary ?Ω,a≥0,b>0 are constants,λ>0 is a real parameter,and p(x)is a continuous function satisfying certain conditions.The existence of two nonnegative nontrivial solutions of the equation is obtained by perturbation method,mountain pass lemma,Ekeland’s variational principle and priori estimate.On the other hand,we study the following p(x)-Kirchhoff fractional equation with concave-convex nonlinearities where Ω(?)RN(N≥3)is a Lipschitz bounded open domain,p,q and r are three continuous bounded functions,λ>0 is a real parameter,1<q(x)<p(x)<r(x)<ps*(x),σp(x,y)(u)=∫Q|u(x)-u(y)|p(x,y)/p(x,y)|x-y|N+sp(x,y)dxdy.By applying the fibering map approach,the existence of two positive solutions of the equation is obtained.