Existence of solutions for modified Kirchhoff-type equation without the Ambrosetti-Rabinowitz condition

Zhongxiang Wang; Gao Jia; Zhongxiang Wang; Gao Jia

doi:10.3934/math.2021272

AIMS Mathematics

2021, Volume 6, Issue 5: 4614-4637. doi: 10.3934/math.2021272

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Research article

Existence of solutions for modified Kirchhoff-type equation without the Ambrosetti-Rabinowitz condition

Zhongxiang Wang ¹,
Gao Jia ^{2
,
,}

1.
Business School, University of Shanghai for Science and Technology, Shanghai, 200093, China
2.
College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

Received: 22 December 2020 Accepted: 04 February 2021 Published: 24 February 2021
MSC : 35J20, 35J60

This paper is devoted to studying a class of modified Kirchhoff-type equations

$ \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\Big)\Delta u+V(x)u-u\Delta(u^2) = f(x,u), \quad \mbox{in}\ \mathbb{R}^3, \end{equation*} $

where $ a > 0, b\geq 0 $ are two constants and $ V:{\mathbb{R}}^{3}\rightarrow {\mathbb{R}} $ is a potential function. The existence of non-trivial solution to the above problem is obtained by the perturbation methods. Moreover, when $ u > 0 $ and $ f(x, u) = f(u) $, under suitable hypotheses on $ V(x) $ and $ f(u) $, we obtain the existence of a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. The character of this work is that for $ f(u)\sim|u|^{p-2}u $ we prove the existence of a positive ground state solution in the case where $ p\in(2, 3] $, which has few results for the modified Kirchhoff equation. Hence our results improve and extend the existence results in the related literatures.
- modified Kirchhoff-type equation,
- ground state solution,
- nehari manifold,
- pohozaev identity
Citation: Zhongxiang Wang, Gao Jia. Existence of solutions for modified Kirchhoff-type equation without the Ambrosetti-Rabinowitz condition[J]. AIMS Mathematics, 2021, 6(5): 4614-4637. doi: 10.3934/math.2021272

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Abstract

This paper is devoted to studying a class of modified Kirchhoff-type equations

$ \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\Big)\Delta u+V(x)u-u\Delta(u^2) = f(x,u), \quad \mbox{in}\ \mathbb{R}^3, \end{equation*} $

where $ a > 0, b\geq 0 $ are two constants and $ V:{\mathbb{R}}^{3}\rightarrow {\mathbb{R}} $ is a potential function. The existence of non-trivial solution to the above problem is obtained by the perturbation methods. Moreover, when $ u > 0 $ and $ f(x, u) = f(u) $, under suitable hypotheses on $ V(x) $ and $ f(u) $, we obtain the existence of a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. The character of this work is that for $ f(u)\sim|u|^{p-2}u $ we prove the existence of a positive ground state solution in the case where $ p\in(2, 3] $, which has few results for the modified Kirchhoff equation. Hence our results improve and extend the existence results in the related literatures.

References

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