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

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
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