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Non-negative solutions of systems of ODEs with coupled boundary conditions
We provide a new existence theory of multiple positive solutions valid for a wide class of
systems of boundary value problems that possess a coupling in the boundary conditions.
Our conditions are fairly general and cover a large number of situations. The theory is illustrated
in details in an example. The approach relies on classical fixed point index
Nonlinear equations involving the square root of the Laplacian
In this paper we discuss the existence and non-existence of weak solutions to
parametric fractional equations involving the square root of the Laplacian
in a smooth bounded domain ()
and with zero Dirichlet boundary conditions. Namely, our simple model is the
following equation \begin{equation*} \left\{ \begin{array}{ll} A_{1/2}u=\lambda
f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on } \partial\Omega.
\end{array}\right. \end{equation*} The existence of at least two non-trivial
-bounded weak solutions is established for large value of the
parameter requiring that the nonlinear term is continuous,
superlinear at zero and sublinear at infinity. Our approach is based on
variational arguments and a suitable variant of the Caffarelli-Silvestre
extension method
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