Geodesic
Ground state solution of a nonlocal boundary-value problem
Electronic journal of differential equations, Tome 2013 (2013)
In this article, we apply the Nehari manifold method to study the Kirchhoff type equation

$ -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u) $

subject to Dirichlet boundary conditions. Under a general 4-superlinear condition on the nonlinearity f, we prove the existence of a ground state solution, that is a nontrivial solution which has least energy among the set of nontrivial solutions. If f is odd with respect to the second variable, we also obtain the existence of infinitely many solutions. Under our assumptions the Nehari manifold does not need to be of class C^1.
Classification : 35J60, 35J25
Keywords: nonlocal problem, Kirchhoff's equation, ground state solution, Nehari manifold 
@article{EJDE_2013__2013__a40,
     author = {Batkam,  Cyril Joel},
     title = {Ground state solution of a nonlocal boundary-value problem},
     journal = {Electronic journal of differential equations},
     year = {2013},
     volume = {2013},
     zbl = {1288.35221},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2013__2013__a40/}
}
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Batkam,  Cyril Joel. Ground state solution of a nonlocal boundary-value problem. Electronic journal of differential equations, Tome 2013 (2013). http://geodesic.mathdoc.fr/item/EJDE_2013__2013__a40/