Volume 54, Issue 4
Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass

Yanghuan Hu, Haidong Liu, Mingjie Wang & Mengjia Xu

J. Math. Study, 54 (2021), pp. 387-395.

Published online: 2021-06

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  • Abstract

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \  u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}

where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.

  • Keywords

Kirchhoff type equation, zero mass, mountain pass approach.

  • AMS Subject Headings

35A15, 35J60

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yanghuan_hu@163.com (Yanghuan Hu)

liuhaidong@mail.zjxu.edu.cn (Haidong Liu)

mingjie_wang_1@163.com (Mingjie Wang)

xumengjia_jx@163.com (Mengjia Xu)

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@Article{JMS-54-387, author = {Yanghuan and Hu and yanghuan_hu@163.com and 16773 and College of Data Science, Jiaxing University, Jiaxing 314001, China and Yanghuan Hu and Haidong and Liu and liuhaidong@mail.zjxu.edu.cn and 16774 and College of Data Science, Jiaxing University, Jiaxing 314001, China and Haidong Liu and Mingjie and Wang and mingjie_wang_1@163.com and 16775 and College of Data Science, Jiaxing University, Jiaxing 314001, China and Mingjie Wang and Mengjia and Xu and xumengjia_jx@163.com and 16778 and College of Data Science, Jiaxing University, Jiaxing 314001, China and Mengjia Xu}, title = {Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass}, journal = {Journal of Mathematical Study}, year = {2021}, volume = {54}, number = {4}, pages = {387--395}, abstract = {

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \  u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}

where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v54n4.21.04}, url = {http://global-sci.org/intro/article_detail/jms/19290.html} }
TY - JOUR T1 - Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass AU - Hu , Yanghuan AU - Liu , Haidong AU - Wang , Mingjie AU - Xu , Mengjia JO - Journal of Mathematical Study VL - 4 SP - 387 EP - 395 PY - 2021 DA - 2021/06 SN - 54 DO - http://doi.org/10.4208/jms.v54n4.21.04 UR - https://global-sci.org/intro/article_detail/jms/19290.html KW - Kirchhoff type equation, zero mass, mountain pass approach. AB -

Consider the Kirchhoff type equation \begin{equation}\label{eq0.1}-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\,dx\right) \Delta u=\left(\frac{1}{|x|^\mu}*F(u)\right)f(u)\ \ \mbox{in}\ \mathbb{R}^N, \ \  u\in D^{1,2}(\mathbb{R}^N), ~~~~~~(0.1)\end{equation}

where $a>0$, $b\geq0$, $0<\mu<\min\{N, 4\}$ with $N\geq 3$, $f: \mathbb{R}\to\mathbb{R}$ is a continuous function and $F(u)=\int_0^u f(t)\,dt$. Under some general assumptions on $f$, we establish the existence of a nontrivial spherically symmetric solution for problem (0.1). The proof is mainly based on mountain pass approach and a scaling technique introduced by Jeanjean.

Yanghuan Hu, Haidong Liu, Mingjie Wang & Mengjia Xu. (2021). Nontrivial Solution for a Kirchhoff Type Problem with Zero Mass. Journal of Mathematical Study. 54 (4). 387-395. doi:10.4208/jms.v54n4.21.04
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