Solutions to Kirchhoff equations with combined nonlinearities

Abstract

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.

Keywords

• Kirchhoff equation
• asymptotically linear
• asymptotically 3-linear
• positive solution
• mountain pass lemma