Multi-bump Solutions for the Quasilinear Choquard Equation in RN

This paper deal with the following quasilinear Choquard equation in R-N : {-delta u - u delta u+ (lambda V(x) + 1)u = ( 1/ |x|(mu )*|u|(p-2 )u, x is an element of R-N, u is an element of H-1 (R-N), where 0 <mu < min{2, 8 - 2N} , 2 <= N < 4 , p is an element of [4 ,2N-mu/N-2), and lambda > 0 is a real parameter. Here, 2(& lowast; )= 2N/ N-2 if N >= 3, 2(& lowast;) = +infinity if N = 2. The potential V: R-N -> R is a nonnegative continuous function verifying some assumptions. Using variational methods, we show that if omega :=int V-1 (0) has several isolated connected components omega(1),... , omega(k )satisfying the interior of omega(j) is non-empty and that & part;omega(j) is smooth, thus for lambda > 0 large enough, the above equation has at least 2(k) -1 multi-bump solutions.