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Existence of Solutions for a Class of Quasilinear Choquard Equations with Potential Vanishing at Infinity

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Abstract

In this paper, we discuss the following quasilinear Choquard equation

$$\begin{aligned} - \Delta v+ V(x)v- v\Delta (v^{2})= \left( \frac{1}{|x|^{\mu }}*Q(x)F(v)\right) Q(x)f( v),~~~~~v\in D^{1,2}(\mathbb {R}^{N}), \end{aligned}$$

where \(N\ge 3\), \(\mu \in (0,N)\), V and Q are differentiable, \(f\in \mathbb {C}(\mathbb {R},\mathbb {R})\) and \(F(s)=\int ^{s}_{0}f(t)dt\). We prove that the above equation has positive solutions with potential vanishing at infinity. When \(Q\equiv 1\) we obtain the existence of nontrivial solution under the general “Berestycki-Lions assumptions” on the nonlinearity f. Based on the result of this case, we also get the existence of nontrivial solutions in the zero mass case by using perturbation method.

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Acknowledgements

The authors are grateful to the anonymous referees for carefully reading this paper and giving valuable comments and advices.

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  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Die Hu & Xianhua Tang

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  1. Die Hu
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  2. Xianhua Tang
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Correspondence to Xianhua Tang.

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Hu, D., Tang, X. Existence of Solutions for a Class of Quasilinear Choquard Equations with Potential Vanishing at Infinity. Bull Braz Math Soc, New Series 54, 31 (2023). https://doi.org/10.1007/s00574-023-00347-7

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