Existence results of positive solutions of Kirchhoff type problems. (English) Zbl 1175.35038

Cheng, Bitao; Wu, Xian

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 10, 4883-4892 (2009).

Summary: In the present paper, we use variational methods to prove two existence results of positive solutions of the following Kirchhoff type problems
\[ \begin{aligned} -\bigg(a+b \int_\Omega |\nabla u|^2\bigg) \Delta u=f(x,u), &\quad\text{in }\Omega;\\ u=0, &\quad\text{on }\partial\Omega. \end{aligned} \]
One deals with the asymptotic behaviors of \(f\) near zero and infinity and the other deals with 4-superlinear of \(f\) at infinity.

MSC:

35J20	Variational methods for second-order elliptic equations
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
49J40	Variational inequalities

Keywords:

critical point; positive solution; Kirchhoff type problems; Sobolev inequality

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