Taiwanese Journal of Mathematics

Two Positive Solutions for Kirchhoff Type Problems with Hardy-Sobolev Critical Exponent and Singular Nonlinearities

Yu-Ting Tang, Jia-Feng Liao, and Chun-Lei Tang

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We consider the following singular Kirchhoff type equation with Hardy-Sobolev critical exponent \[ \begin{cases} \displaystyle -\left( a + b \int_{\Omega} |\nabla u|^2 \, dx \right) \Delta u = \frac{u^{3}}{|x|} + \frac{\lambda}{|x|^{\beta} u^{\gamma}}, & x \in \Omega, \\ u > 0, & x \in \Omega, \\ u = 0, & x \in \partial \Omega, \end{cases} \] where $\Omega \subset \mathbb{R}^{3}$ is a bounded domain with smooth boundary $\partial \Omega$, $0 \in \Omega$, $a,b,\lambda \gt 0$, $0 \lt \gamma \lt 1$, and $0 \leq \beta \lt (5+\gamma)/2$. Combining with the variational method and perturbation method, two positive solutions of the equation are obtained.

Article information

Taiwanese J. Math., Volume 23, Number 1 (2019), 231-253.

Received: 10 December 2017
Revised: 27 May 2018
Accepted: 23 July 2018
First available in Project Euclid: 1 August 2018

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Zentralblatt MATH identifier

Primary: 35A15: Variational methods 35B33: Critical exponents 35J75: Singular elliptic equations

Kirchhoff type equation Hardy-Sobolev critical exponent singularity positive solution variational method


Tang, Yu-Ting; Liao, Jia-Feng; Tang, Chun-Lei. Two Positive Solutions for Kirchhoff Type Problems with Hardy-Sobolev Critical Exponent and Singular Nonlinearities. Taiwanese J. Math. 23 (2019), no. 1, 231--253. doi:10.11650/tjm/180705. https://projecteuclid.org/euclid.twjm/1533110480

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