Some Existence Results on a Class of Generalized Quasilinear Schrödinger Equations with Choquard Type

Abstract

In this paper, we study the generalized quasilinear Schrödinger equation

$$\begin{aligned} -\text {div}(g^2(u)\nabla u)+g(u)g'(u)|\nabla u|^2+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,\ \ \ x\in {{\mathbb {R}}}^N, \end{aligned}$$

where \(N\ge 3\), \(0<\alpha <N\), \(\frac{2(N+\alpha )}{N}< p<\frac{2(N+\alpha )}{N-2}\), \(V: {{\mathbb {R}}}^N\rightarrow {{\mathbb {R}}}\) is a potential function and \(I_{\alpha }\) is a Riesz potential. Under appropriate assumptions on g and V(x), we establish the existence of positive solutions and ground state solutions.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11661053, 11771198, 11961045, and 11901276), the Provincial Natural Science Foundation of Jiangxi (Grant Nos. 20161BAB201009, 20181BAB201003, 20202BAB201001 and 20202BAB211004), the Outstanding Youth Scientist Foundation Plan of Jiangxi (Grant No. 20171BCB23004).

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Correspondence to Xianjiu Huang.

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Ling, P., Huang, X. & Chen, J. Some Existence Results on a Class of Generalized Quasilinear Schrödinger Equations with Choquard Type. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-021-00585-9

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Keywords

  • Quasilinear Schrödinger equation
  • Positive solutions
  • Ground state solutions
  • Choquard type

Mathematics Subject Classification

  • 35J60
  • 35J20