R^N上临界增长p-Kirchhoff型方程的非平凡解的存在性

The existence of nontrivial solutions for p-Kirchhoff type equations with critical exponent in R^N

本文主要研究以下具临界增长的非线性p-Kirchhoff型方程的非平凡解的存在性:{-(a+b∫_(R^N)|▽u|~p)?_pu=|u|^(p*-2)u+μf (x)|u|^(q-2)u, x∈R^N,(0.1) u∈D^(1,p)(R^N),其中a≥0,b>0,1<p<N,1<q<p,p*=N_p/(N-p),μ≥0,?_pu=div(|▽u|^(p-2)▽u)表示p-Laplace算子对函数u的作用, f∈L(p*/(p*-q))(R^N)\{0}且f是非负的.本文利用Ekeland变分原理和山路定理证明方程(0.1)在适当条件下至少存在两个非平凡解.

We study the existence of nontrivial solutions of the p-Kirchhoff type equation-(a+b/R^N|▽μ|p)Δpu=|μ|p*-2u+μf(x)|μ|q-2μ,x∈GN,μ∈D1,p(R^N)with critical nonlinearity,where a≥O,b>Q,l<p<N,l<q<p,p*=Mp/N-p,Δpμ=div(|▽μ|p-2▽μ)is the p-Laplacian off∈Lp*/p*-q(R^N)\{0}and f is nonnegative.Under certain conditions,we prove the existence of at least two nontrivial solutions of the above equation,by the methods of Ekeland variational principle and the mountain pass theorem.