Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type. (English) Zbl 1352.34025
The authors are concerned with the Kirchoff-type equation
\[
K\left(\int_a^b\mid u'(x)\mid^2dx\right)u''=\lambda f(x,u),\quad x\in(a,b)
\]
with homogeneous Dirichlet boundary conditions. \(\lambda\) is a positive real parameter, \(K: [0,+\infty)\to\mathbb{R}\) is continuous and lower bounded while the nonlinearity \(f\) is \(L^1\)-Carathéodory. Using variational techniques, the authors prove the existence of a weak solution in the Sobolev space \(W_0^{1,2}\) for some values of the parameter \(\lambda\).
Reviewer: Smail Djebali (Algiers)
MSC:
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 47J10 | Nonlinear spectral theory, nonlinear eigenvalue problems |
| 58E50 | Applications of variational problems in infinite-dimensional spaces to the sciences |
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