Taiwanese Journal of Mathematics

Ground State Solutions for Kirchhoff-type Problems with Critical Nonlinearity

Yiwei Ye

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In this paper, we study the Kirchhoff-type equation with critical exponent \[ -\left( a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx \right) \Delta u + V(x)u = a(x) f(u) + u^5 \quad \textrm{in $\mathbb{R}^3$}, \] where $a,b \gt 0$ are constants, $V \in C(\mathbb{R}^3,\mathbb{R})$, $\lim_{|x| \to \infty} V(x) = V_{\infty} \gt 0$ and $V(x) \leq V_{\infty} + C_1 e^{-b |x|}$ for some $C_1 \gt 0$ and $|x|$ large enough. Via variational methods, we prove the existence of ground state solution.

Article information

Taiwanese J. Math., Volume 24, Number 1 (2020), 63-79.

Received: 2 August 2017
Accepted: 6 March 2019
First available in Project Euclid: 15 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45M20: Positive solutions
Secondary: 35B33: Critical exponents

Kirchhoff-type equation ground state solutions variational methods


Ye, Yiwei. Ground State Solutions for Kirchhoff-type Problems with Critical Nonlinearity. Taiwanese J. Math. 24 (2020), no. 1, 63--79. doi:10.11650/tjm/190402. https://projecteuclid.org/euclid.twjm/1555315213

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