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 where Δ p is the p -Laplacian operator, 1 < p < N , V is a continuous real function on ℝ N , 0 < μ < N , F ( s ) is the primitive function of f ( s ), ε 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.