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Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent. (English) Zbl 1398.35094

Summary: We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -\Delta u + u = (I_\alpha \ast | u |^{\frac{\alpha}{N} + 1}) | u |^{\frac{\alpha}{N} - 1} u + f(x, u)\quad \text{ in }\, \mathbb{R}^N \] where \(N \geq 1\), \(I_\alpha\) is the Riesz potential of order \(\alpha \in(0, N)\), the exponent \(\frac{\alpha}{N} + 1\) is critical with respect to the Hardy-Littlewood-Sobolev inequality and the nonlinear perturbation \(f\) satisfies suitable growth and structural assumptions.

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
35J20 Variational methods for second-order elliptic equations
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