Solvability Of A Type Of Kirchhoff Equation With Variable Exponent

In this paper,we study the solvability of a class of Kirchhoff equations with variable exponent by using Ekeland’s variational principle,mountain pass lemma,invariant sets for the descending flow and other variational methods,perturbation method and iterative technique.On the one hand,this paper considers the following Kirchhoff type equations with variable exponent (?) where Ω(?) R3 is a bounded domain with smooth boundary ?Ω,a≥0,b>0 are two constants,q(x)∈ C((?))and 4≤q(x)≤(?)q(x)=q+<6 for x ∈ (?).By using perturbation method,mountain pass lemma and a priori estimation,a nonnegative nontrivial solution of the equation is obtained,and infinitely many sign-changing solutions of the equation are obtained by using the method of invariant sets for the descending flow.On the other hand,this paper investigates the following Kirchhoff type equations with variable exponent(?) where B is the unit ball in R3,a≥ 0,b>0 are two constants,q(x)=q(|x|)∈C((?)),and 2<(?)q(x)=q-<4<q+=(?)q(x)<6,q(0)>4.By using perturbation method,mountain pass lemma,Ekeland’s variational principle and a priori estimation,the existence and multiplicity of radial solutions of the equation are proved.