## 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

### Keywords:

nonlinear Choquard equations; ground state solution
Full Text:

### References:

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