Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method. (English) Zbl 1366.35050

The authors consider the quasilinear Choquard equation, which involves the \(p\)-Laplacian operator and a general nonlinear term (instead of the Laplacian and a power-type nonlinear term). Under the hypothesis that the potential function has a local minimum, they prove the existence, multiplicity and concentration behaviour of positive solutions for the equation. The approach is based on the penalization method and the Lyusternik-Schnirelmann category theory (for proving the multiplicity result) and on the Moser iteration method (for studying the concentration behaviour). The results are new even for the semilinear case \(p=2\).


35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J50 Variational methods for elliptic systems
35J62 Quasilinear elliptic equations
Full Text: DOI