| Abstract: |
| We will discuss the question of $\bf existence $
and $\bf multiplicity $ of positive solutions
to the semilinear elliptic Dirichlet problem
(1)$$
{}- \Delta u = \lambda\, u(x)^{q(x) - 1} + f(x,u(x))
\quad\mbox{ for }\, x\in \Omega \,;\qquad
u = 0
\quad\mbox{ on }\, \partial\Omega \,,
$$
where $\Omega\subset \mathbf{R}^N$ is a bounded domain with
the boundary of class $C^{1,\alpha}$,
$\lambda\in \mathbf{R}^1$ a spectral parameter, and
$f(x,u) = |u|^{r-1}\, u$ is a $\bf signed$ $r$-$\bf power$
($r > 0$)
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 (1)
with a $\bf convex$ and $\bf concave$
non-linear reaction
$
f(x, \,\cdot\,)\colon s\longmapsto |s|^{q(x) - 2} s\colon
\mathbf{R}^1_+\subset \mathbf{R}^1\to \mathbf{R}^1
$
which (for $s\geq 0$) is $\bf convex$
in a nonempty open subset
$
\Omega_{+}\stackrel{{\mathrm{def}}}{=}
\{ x\in \Omega\colon q(x) > 2\}
$
and $\bf concave$ in another nonempty open subset
$
\Omega_{-}\stackrel{{\mathrm{def}}}{=}
\{ x\in \Omega\colon q(x) < 2\}
$
of a bounded domain $\Omega\subset \mathbf{R}^N$.
Here, $\lambda\in \mathbf{R}^1_+$ is a non-negative
spectral parameter which decides about the existence
and multiplicity of positive weak solutions (at least two)
to problem (1) in case we take $f\equiv 0$.
Our main contribution is a method how to handle the interplay between
convex and concave non-linearities in two disjoint
nonempty open subsets of a domain $\Omega$
(connected in $\mathbf{R}^N$),
as opposed to the classical works assuming
a non-linearity $f(s)$ being concave
for small values of $s\in \mathbf{R}^1_+$ and convex
for large $s\in \mathbf{R}^1_{+}$, uniformly in $\Omega$.
Finally, if time permits, we will discuss also the classical question
of $\bf uniqueness$ for a related problem with the
$p(x)$-Laplacian provided
$q(x)\leq \mathrm{const}_1 < \mathrm{const}_2\leq p(x)$
holds for all $x\in \Omega$, and for $p(x)\equiv 2$
the $\bf multiplicity$ of large solution branches
bifurcating from infinity as $\lambda\searrow 0+$. |
|