一类具有临界指数增长的分数阶p-Laplace方程的变分问题

The Variational Problem of a Class of Fractional p-Laplace Equation with Critical Exponential Growth

本文研究了分数阶p-Laplace方程(-Δ)p^s+V(x)u^p-2u=K(x)|u|^q-2u+|u|ps^*-2u(x∈RN),其中,s∈(0,1),ps*=Np/N-sP,N>sp,p>1,并且V(x)和K(x)是正连续函数,(-Δ)p^s是非线性局部p-Laplace算子,定义如下:(-△)p^su(x)-2limε→0∫BE(x)^c|u(x)-u(y)|^p-2(u(x)-u(y))/|x-y|^N+spdy,通过应用集中紧原理、山路定理、Moser迭代等变分方法,给出方程对应的变分框架,并给出对应泛函J(·)的集中紧性以及其(PS)c。序列的收敛性的证明。

In this paper,we concerned with the fractional equation of the form:(-Δ)p^s+V(x)u^p-2u=K(x)|u|^q-2u+|u|ps^*-2u(x∈RN),ps*=Np/N-sP,N>sp,p>1,V(x)and K(x)are non-negative continuous potentials that V(x)may decay to zero as|x|→∞,the definition of the nonlocal p-Laplace operator(-△)ps is:(-△p^su(x)-2limε)→0∫BE(x)^c|u(x)-u(y)|^p-2(u(x)-u(y))/|x-y|^N+spdy,we apply some variational methods such as the principle of concentration、Mountain Pass theorem and iteration method to research the above equation,therefore the variational frame corresponding to the equation is given.Furthermore we can get the proof of the compactness of the corresponding functional set and convergence of the(PS)c sequence.