Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method

We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equation -epsilon(p)Delta(p)u + V(x)vertical bar u vertical bar(p-2)u = epsilon(mu-N)(1/vertical bar x vertical bar(mu) * F(u))f(u) in R-N, where Delta(p) is the p-Laplacian operator, 1 < p < N, V is a continuous real function on R-N, 0 < mu < N, F(s) is the primitive function of f(s), is an element of is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e. V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik-Schnirelmann theory and even show novel results for the semilinear case p = 2.