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Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior. (English) Zbl 1288.35456

Summary: Existence and bifurcation of positive solutions to a Kirchhoff type equation \[ \begin{cases} -\left(a+b\int\limits_{\varOmega}|\nabla u|^2\right)\Delta u=\nu f(x,u),\quad &\text{in }\varOmega ,\\u=0,\quad &\text{on }\partial\varOmega\end{cases} \] are considered by using topological degree argument and variational method. Here \(f\) is a continuous function which is asymptotically linear at zero and is asymptotically 3-linear at infinity. The new results fill in a gap of recent research about the Kirchhoff type equation in bounded domain, and in our results the nonlinearity may be resonant near zero or infinity.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74K05 Strings
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A15 Variational methods applied to PDEs
74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
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