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Some perturbation results of Kirchhoff type equations via Morse theory. (English) Zbl 07254121
Summary: In this paper, we consider the following Kirchhoff type equation: \[ \begin{cases} - (a+b \int_{\varOmega } \vert \nabla u \vert^2\,dx ) \Delta u= f(x,u)& \text{in } \varOmega , \\ u=0 & \text{on } \partial \varOmega , \end{cases}\] where \(a,b>0\) are constants and \(\varOmega \subset \mathbb{R}^N (N=1,2,3)\) is a bounded domain with smooth boundary \(\partial \Omega \). By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either \(a\) or \(b\) being sufficiently small.
35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
Full Text: DOI
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