Infinitely Many Solutions for Critical Degenerate Kirchhoff Type Equations Involving the Fractional p-Laplacian

In this paper we study a class of critical Kirchhoff type equations involving the fractional p-Laplacian operator, that is M R2N | u( x) - u( y)| p | x - y| N+ ps dxdy (- ) s pu=.w( x)| u| q- 2u + | u| p * s - 2u, x. RN where ps is the fractional p-Laplacian operator with 0<s<1<p<infinity, dimension N>ps, 1<q<ps, ps is the critical exponent of the fractional Sobolev space Ws,p(RN), lambda is a positive parameter, M is a non-negative function while w is a positive weight. By exploiting Kajikiya's new version of the symmetric mountain pass lemma, we establish the existence of infinitely many solutions which tend to zero under a suitable value of lambda. The main feature and difficulty of our equations is the fact that the Kirchhoff term M is zero at zero, that is the equation is degenerate. To our best knowledge, our results are new even in the Laplacian and p-Laplacian cases.