| Contents |
| In this contribution we study nonlinear elliptic problems with a
singular term of the form
\begin{equation} \label{eq:1}
\left\{\begin{array}{cc} -\mbox{div}\; a (Du(z))
=\beta(z)u(z)^{-\gamma} +
f(z,u(z)), \quad \mbox{for almost all } \;z \in \Omega, \\ \\ u \mid_{\partial
\Omega}=0, \quad \gamma \in (0, 1), \end{array} \right.
\end{equation}
where $\Omega \subseteq \mathbb{R}^N\;(N \geq 1)\;$ is a bounded
domain with a $C^2$-boundary $\partial \Omega, \;a: \mathbb{R}^N
\rightarrow \mathbb{R}^N$ is strictly monotone with certain
regularity properties and $\beta \in C(\Omega) \cap
L^{\infty}(\Omega)_{+} \!\setminus \{0\}\;$. Moreover, $\;f\in
C(\Omega \times \mathbb{R})\;$ being either superlinear near
$+\infty$ (without satisfying the Ambrosetti-Rabinowitz condition)
or sublinear near
$+\infty$.
We prove three multiplicity theorems producing at least two smooth
positive solutions for problems of the above form. In the first one, the leading
differential operator is not in general homogeneous and the
result is obtained for ``small'' $\;||b||_{\infty}\;$. The other
two results concern problems driven by
the $p$-Laplacian differential operator. In these cases, no
restrictions are imposed on $\;||b||_{\infty}\;$.
Our approach is variational employing suitable truncation and
comparison techniques. |
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