## Rocky Mountain Journal of Mathematics

### Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$

#### Abstract

We study the existence of solutions for a class of Kirchhoff type problems with critical growth in $\mathbb {R}^N$:$$-\varepsilon ^2\biggl (a+b\int _{\mathbb {R}^N}|\nabla u|^2\,dx\biggr )\Delta u + V(x)u -\varepsilon ^2a\Delta (u^2)u = |u|^{22^\ast -2}u + h(x,u),$$ $(t, x) \in \mathbb {R} \times \mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We prove that it has at least one solution and for any $m \in \mathbb {N}$, it has at least $m$ pairs of solutions. The proofs are based on the variational methods and concentration-compactness principle.

#### Article information

Source
Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1725-1753.

Dates
First available in Project Euclid: 19 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1568880099

Digital Object Identifier
doi:10.1216/RMJ-2019-49-5-1725

Mathematical Reviews number (MathSciNet)
MR4010581

#### Citation

Zhang, Jing; Chen, Alatancang. Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$. Rocky Mountain J. Math. 49 (2019), no. 5, 1725--1753. doi:10.1216/RMJ-2019-49-5-1725. https://projecteuclid.org/euclid.rmjm/1568880099

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