## 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 $$p=2$$.

### MSC:

 35J92 Quasilinear elliptic equations with $$p$$-Laplacian 35J50 Variational methods for elliptic systems 35J62 Quasilinear elliptic equations
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