Positive solutions for a Kirchhoff type problem with fast increasing weight and critical nonlinearity
In this paper, we study the following Kirchhoff type problem \[ -\left(a+b\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\right)\hbox{div}(K(x)\nabla u)=\lambda K(x)|x|^{\beta}|u|^{q-2}u+K(x)|u|^{4}u,\quad x\in \mathbb{R}^3, \] where $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$, $\beta=(\alpha-2)(6-q)...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2019-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=7402 |
| Summary: | In this paper, we study the following Kirchhoff type problem
\[
-\left(a+b\int_{\mathbb{R}^3} K(x)|\nabla u|^2dx\right)\hbox{div}(K(x)\nabla u)=\lambda K(x)|x|^{\beta}|u|^{q-2}u+K(x)|u|^{4}u,\quad x\in \mathbb{R}^3,
\]
where $K(x)=\exp({|x|^{\alpha}/4})$ with $\alpha\geq2$, $\beta=(\alpha-2)(6-q)/4$ and the parameters $a, b, \lambda >0$. When $6-\frac{4}{\alpha}<q<6$, we obtain a positive ground state solution for any $\lambda>0$. When $2<q<4$, we obtain a positive solution for $\lambda>0$ small enough. In the proof we use variational methods. |
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| ISSN: | 1417-3875 |