| Contents |
| Let $\Omega$ be a nonempty connected bounded open set in
$\mathbb{R}^N$ with smooth boundary $\partial \Omega$. Moreover, let
$s\in ]1,2[$ and $r\in ]1,s[$. We present some existence and
multiplicity results of nonzero and nonnegative solutions for the
following elliptic problem involving a nonlinearity indefinite in
sing
\begin{eqnarray*}
\left\{\begin{array}{lll}
-\Delta u=\lambda u^{s-1}-u^{r-1} \ \ \ \ &{\rm in} \ \ \ \
&\Omega,\\
u=0 & {\rm on} & \partial \Omega.
\end{array}\right.
\end{eqnarray*}
Here, $\lambda$ is a positive parameter. Particular emphasis is
devoted to the question of finding positive solutions to the above
problem. The main difficulty, in dealing with this question, comes
from the particular structure of the right hand side which prevents
us to use the classical Strong Maximum Principle in order to obtain
the positivity of every nonzero and nonnegative solution. Further
results related to the singular case $r\in ]0,1[$ are also presented
jointly to some open problems concerning both the singular and
non-singular case. |
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