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Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities

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  • In this work the authors present some existence, non-existence and location results of the problem composed of the fourth order fully nonlinear equation \begin{equation*} u^{\left( 4\right) }\left( x\right) +f( x,u\left( x\right) ,u^{\prime }\left( x\right) ,u^{\prime \prime }\left( x\right) ,u^{\prime \prime \prime }\left( x\right) ) =s\text{ }p(x) \end{equation*} for $x\in \left[ a,b\right] ,$ where $f:\left[ a,b\right] \times \mathbb{R} ^{4}\rightarrow \mathbb{R},$ $p:\left[ a,b\right] \rightarrow \mathbb{R}^{+}$ are continuous functions and $s$ a real parameter, with the boundary conditions \begin{equation*} u\left( a\right) =A,\text{ }u^{\prime }\left( a\right) =B,\text{ }u^{\prime \prime \prime }\left( a\right) =C,\text{ }u^{\prime \prime \prime }\left( b\right) =D,\text{ } \end{equation*} for $A,B,C,D\in \mathbb{R}.$ In this work they use an Ambrosetti-Prodi type approach, with some new features: the existence part is obtained in presence of nonlinearities not necessarily bounded, and in the multiplicity result it is not assumed a speed growth condition or an asymptotic condition, as it is usual in the literature for these type of higher order problems.
        The arguments used apply lower and upper solutions technique and topological degree theory.
        An application is made to a continuous model of the human spine, used in aircraft ejections, vehicle crash situations, and some forms of scoliosis.
    Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18, 47H11.


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