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Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent
Published online by Cambridge University Press: 20 November 2018
Abstract
In this paper, we consider the following critical Kirchhoff type equation:
$$\left\{ _{u\,=\,0,\,\,\,\,\,\,\text{on}\,\partial \Omega \text{,}}^{-(a\,+\,b{{\int }_{\Omega }}|\nabla u{{|}^{2}})\Delta u\,=\,\text{Q(}x)|u{{|}^{4}}u\,+\,\lambda |u{{|}^{q-1}}u,\,\,\,\text{in}\,\Omega \text{,}} \right.$$
By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\,<\,q\,<\,5$. The relation between the number of maxima of
$\text{Q}$ and the number of positive solutions for the problem is also investigated.
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- Copyright © Canadian Mathematical Society 2018
Footnotes
This work is partially supported by the China Scholarship Council, NSFC (Nos: 11626202, 11571370, 11471278, 11601525) and the Hunan Provincial Innovation Foundation for Postgraduates (CX2015B037).
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