Solutions to Kirchhoff equations with combined nonlinearities
We prove the existence of multiple positive solutions for the Kirchhoff equation $$\displaylines{ -\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =h(x)u^q+f(x,u), \quad x\in \Omega, \cr u=0, \quad x\in\partial \Omega, }$$ Here $\Omega $ is an open bounded domain in $ R^{N}$ ($N=1,2,3$), $...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2014-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2014/10/abstr.html |
| Summary: | We prove the existence of multiple positive solutions for the
Kirchhoff equation
$$\displaylines{
-\Big(a+b\int_{\Omega}|\nabla u|^2dx\Big)\Delta u =h(x)u^q+f(x,u), \quad
x\in \Omega, \cr
u=0, \quad x\in\partial \Omega,
}$$
Here $\Omega $ is an open bounded domain in $ R^{N}$ ($N=1,2,3$),
$h(x)\in L^\infty(\Omega)$, $f(x,s)$ is a continuous function which
is asymptotically linear at zero and is asymptotically 3-linear at infinity.
Our main tools are the Ekeland's variational principle and the
mountain pass lemma. |
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| ISSN: | 1072-6691 |