### On a second order boundary value problem with singular nonlinearity

DOI: http://dx.doi.org/10.12775/TMNA.2006.001

#### 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 $E$.

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 $E$.

#### Keywords

Variational methods; elliptic problems; singular nonlinearity

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