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2019 Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$
Jing Zhang, Alatancang Chen
Rocky Mountain J. Math. 49(5): 1725-1753 (2019). DOI: 10.1216/RMJ-2019-49-5-1725

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.

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Jing Zhang. Alatancang Chen. "Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$." Rocky Mountain J. Math. 49 (5) 1725 - 1753, 2019. https://doi.org/10.1216/RMJ-2019-49-5-1725

Information

Published: 2019
First available in Project Euclid: 19 September 2019

MathSciNet: MR4010581
Digital Object Identifier: 10.1216/RMJ-2019-49-5-1725

Subjects:
Primary: 58E05
Secondary: 58E50

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.49 • No. 5 • 2019
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