## Existence and qualitative properties of solutions for Choquard equations with a local term.(English)Zbl 1412.35123

Summary: In this paper, a nontrivial solution $$u \in H^1(\mathbb{R}^N)$$ to the autonomous Choquard equation with a local term $- \Delta u + \lambda u = (I_\alpha \ast | u |^p) | u |^{p - 2} u + | u |^{q - 2} u \quad\text{in } \mathbb{R}^N$ is obtained, where $$N \geq 3$$, $$\alpha \in(0, N)$$, $$\lambda > 0$$ is a constant, $$I_\alpha$$ is the Riesz potential, $$\frac{N + \alpha}{N} < p < \frac{N + \alpha}{N - 2}$$ and $$q \in(2, 2^\ast = \frac{2 N}{N - 2})$$. Under some further assumptions on $$p$$ and $$q$$, the regularity and the Pohožaev identity of the solution are established, and then it is shown that the obtained solution is a groundstate of mountain pass type. Moreover, the positivity and symmetry of the groundstate are also considered. By using the results obtained for the autonomous equation, a positive groundstate solution for the nonautonomous equation $- \Delta u + V(x) u = (I_\alpha \ast | u |^p) | u |^{p - 2} u + | u |^{q - 2} u \quad\text{in } \mathbb{R}^N$is also found under some assumptions on $$V(x)$$.

### MSC:

 35J60 Nonlinear elliptic equations 35B09 Positive solutions to PDEs
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### References:

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