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Semi-classical states for the Choquard equation. (English) Zbl 1309.35029

The authors use a newly developed penalization technique to establish the existence of solutions for semilinear equations in the semi-classical setting. The nonlinearity on the right hand side is non-local as it involves convolution with a Riesz potential. Assuming the external potential \(V\) has a local minimum and under certain decay conditions, the authors prove the existence of a family of solutions which concentrates at the local minimum of \(V\) as the semi-classical parameter \(\varepsilon\) tends to zero.

MSC:

35J61 Semilinear elliptic equations
35B09 Positive solutions to PDEs
35B25 Singular perturbations in context of PDEs
35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
45K05 Integro-partial differential equations
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