On a second order boundary value problem with singular nonlinearity

Abstract

In this paper we investigate in a variational setting, the elliptic
boundary value problem $-\Delta u={\textt{sign}u}/{|u|^{\alpha+1}}$ in
$\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an open connected
bounded subset of ${\mathbb R}^N$, and $\alpha> 0$. For the positive solution,
which is checked as a minimum point of the formally associated
functional
$$
E(u)=\frac 12\int_\Omega|\nabla u|^2+\frac{1}{\alpha}
\int_\Omega \frac1{|u|^\alpha},
$$
we prove dependence
on the domain $\Omega$. Moreover, an approximative functional $E_\varepsilon$ is
introduced, and an upper bound for the sequence of mountain pass points
$u_\varepsilon$ of $E_\varepsilon$, as $\varepsilon\to 0$, is given. For the onedimensional
case, all sign-changing solutions of
$-u''={\text{sign}u}/{|u|^{\alpha+1}}$ are characterized by their
nodal set as the mountain pass point and $n$-saddle points ($n> 1$) of
the functional $E$.