Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. (English) Zbl 1160.35421

Mao, Anmin; Zhang, Zhitao

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 3, 1275-1287 (2009).

From the summary: We use minimax methods and invariant sets of descent flow to prove two existence theorems for the 4-superlinear Kirchhoff type problems
\[ \begin{aligned} -\bigg(a+b \int_\Omega |\nabla u|^2\bigg)\Delta u= f(x,u) &\quad\text{in }\Omega,\\ u=0 &\quad\text{on }\partial\Omega, \end{aligned} \]
without the P.S. condition, one concerning the existence of a nontrivial solution and the other one concerning the existence of sign-changing solutions and multiple solutions.

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J20	Variational methods for second-order elliptic equations
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems

Keywords:

critical points; sign-changing solutions; multiple solutions; Kirchhoff type nonlocal problems

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