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Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros

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  • We study the multiplicity solutions for the nonlinear elliptic equation

    $ -\mathcal{M}_{\lambda,\Lambda}^+ (D^2u)=f(u) $ in $\Omega$,

    $ u=0 $ on $\partial \Omega$

    and a more general fully nonlinear elliptic equation

    $ F(D^2u)=f(u) $ in $\Omega$,

    $ u=0 $ on $\partial \Omega$,

    where $\Omega$ is a bounded domain in $\mathbb{R}^N, N\geq 3$, $f$ is a locally Lipschitz continuous function with superlinear growth at infinity. We will show that the equation has at least two positive solutions under some assumptions.

    Mathematics Subject Classification: Primary: 35J60, 35J65; Secondary: 35B45.


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