On a class of nonlocal elliptic problems with critical growth. (English) Zbl 1198.35281

Alves, C. O.; Corrêa, F. J. S. A.; Figueiredo, G. M.

Differ. Equ. Appl. 2, No. 3, 409-417 (2010).

Summary: This paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of Kirchhoff type
\[ -\left[M\left(\int_{\Omega} |\nabla u|^2\, dx \right)\right]\Delta u= \lambda f(x,u)+ u^5 \;\text{ in }\Omega,\quad u(x)>0\;\text{ in }\Omega \quad\text{and}\quad u=0\;\text{ on }\partial \Omega , \]
where \(\Omega\subset\mathbb R^N\), for \(N=1,2\) and 3, is a bounded smooth domain, \(M\) and \(f\) are continuous functions and \(\lambda\) is a positive parameter. Our approach is based on the variational method.

MSC:

35R09	Integro-partial differential equations
45K05	Integro-partial differential equations
35A15	Variational methods applied to PDEs
35J25	Boundary value problems for second-order elliptic equations
35B09	Positive solutions to PDEs

Keywords:

variational methods; nonlocal problems; Kirchhoff equation; critical growth