Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent

Qin, Dongdong ; He, Yubo ; Tang, Xianhua

Canadian mathematical bulletin, Tome 61 (2018) no. 2, pp. 353-369

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-041-x

Mots-clés : 35J20, 35J60, 35J25, Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state

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