Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities
Feliz Minhós - Departamento de Matemática. Universidade de Évora, Centro de Investigação em Matemática e Aplicaçoes da U.E. (CIMA-UE), Rua Romão Ramalho, 59. 7000-671 Évora, Portugal (email) Abstract:
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.
Keywords: Higher order Ambrosetti-Prodi problems, one-sided Nagumo condition, existence, non-existence and multiplicity of solutions.
Received: September 2012; Revised: April 2013; Available Online: November 2013. |