Existence results of positive solutions of Kirchhoff type problems. (English) Zbl 1175.35038
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.
\[ \begin{aligned} -\bigg(a+b \int_\Omega |\nabla u|^2\bigg) \Delta u=f(x,u), &\quad\text{in }\Omega;\\ u=0, &\quad\text{on }\partial\Omega. \end{aligned} \]
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 |
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