Geodesic
Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent
Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 353-369

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DOI

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.
DOI : 10.4153/CMB-2017-041-x
Mots-clés : 35J20, 35J60, 35J25, Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state 
Qin, Dongdong; He, Yubo; Tang, Xianhua. Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent. Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 353-369. doi: 10.4153/CMB-2017-041-x
@article{10_4153_CMB_2017_041_x,
     author = {Qin, Dongdong and He, Yubo and Tang, Xianhua},
     title = {Ground {State} and {Multiple} {Solutions} for {Kirchhoff} {Type} {Equations} {With} {Critical} {Exponent}},
     journal = {Canadian mathematical bulletin},
     pages = {353--369},
     year = {2018},
     volume = {61},
     number = {2},
     doi = {10.4153/CMB-2017-041-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-041-x/}
}
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%J Canadian mathematical bulletin
%D 2018
%P 353-369
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