| Abstract: |
| We will discuss the question of {\it\bfseries existence\/}
and {\it\bfseries multiplicity\/} of {\it positive solutions\/}
to the semilinear elliptic Dirichlet problem
%
\begin{equation}
\label{e:semi-lin}
\begin{alignedat}{3}
{}- \Delta u = \lambda\, u(x)^{q(x) - 1} + f(x,u(x))
\quad\mbox{ for }\, x\in \Omega \,;\qquad
u = 0
\quad\mbox{ on }\, \partial\Omega \,,
\end{alignedat}
\end{equation}
%
where $\Omega\subset \RR^N$ is a bounded domain with
the boundary of class $C^{1,\alpha}$,
$\lambda\in \RR$ a spectral parameter, and
$f(x,u) = |u|^{r-1}\, u$ is a \underline{\bf signed $r$\--power}
($r > 0$)
of the unknown function of (a positive variable) $u\in (0,\infty)$
which depends on the point $x\in \Omega$; $r = q(x) - 1$, for instance.
We will briefly present basic methods for treating
the semilinear problem \eqref{e:semi-lin}
with a {\it\bfseries convex\/} and {\it\bfseries concave\/}
non\-linear reaction
%
\begin{math}
f(x, \,\cdot\,)\colon s\longmapsto |s|^{q(x) - 2} s\colon
\RR_+\subset \RR\to \RR
\end{math}
%
which (for $s\geq 0$) is {\it\bfseries convex\/}
in a nonempty open subset
%
\begin{math}
\Omega_{+}\eqdef \{ x\in \Omega\colon q(x) > 2\}
\end{math}
%
and {\it\bfseries concave\/} in another nonempty open subset
%
\begin{math}
\Omega_{-}\eqdef \{ x\in \Omega\colon q(x) < 2\}
\end{math}
%
of a bounded domain $\Omega\subset \mathbb{R}^N$.
Here, $\lambda\in \RR_+$ is a non\-negative
spectral parameter which decides about the existence
and multiplicity of positive weak solutions (at least two)
to problem \eqref{e:semi-lin} in case we take $f\equiv 0$.
Our main contribution is a method how to handle the interplay between
{\it convex\/} and {\it concave\/} non\-linearities in two disjoint
nonempty open subsets of a domain $\Omega$ (connected in $\RR^N$),
as opposed to the classical works assuming
a non\-linearity $f(s)$ being {\it concave\/}
for small values of $s\in \RR_+$ and {\it convex\/}
for large $s\in \RR_{+}$, uniformly in $\Omega$. |
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