First we recall a Faber-Krahn type inequality and an estimate for ${\lambda }_p\left(\Omega \right)$ in terms of the so-called Cheeger constant. Then we prove that the eigenvalue ${\lambda }_p\left(\Omega \right)$ converges to the Cheeger constant $h\left(\Omega \right)$ as $p\to 1$. The associated eigenfunction $u_p$ converges to the characteristic function of the Cheeger set, i.e. a subset of $\Omega$ which minimizes the ratio $\left|\partial D\right|/\left|D\right|$ among all simply connected $D\subset \subset \Omega$. As a byproduct we prove that for convex $\Omega$ the Cheeger set $\omega$ is also convex.

We study the Dirichlet boundary value problem for the $p$-Laplacian of the form $$-{\Delta }_{p}u-{\lambda }_1{\left|u\right|}^{p-2}u=f\text{in}\Omega ,u=0\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$ is a bounded domain with smooth boundary $\partial \Omega$, $N\ge 1$, $p>1$, $f\in C\left(\overline{\Omega }\right)$ and ${\lambda }_1>0$ is the first eigenvalue of ${\Delta }_p$. We study the geometry of the energy functional $$E_p\left(u\right)=\frac{1}{p}{\int }_{\Omega }{\left|\nabla u\right|}^p-\frac{{\lambda }_1}{p}{\int }_{\Omega }{\left|u\right|}^p-{\int }_{\Omega }fu$$ and show the difference between the case $1<p<2$ and the case $p>2$. We also give the characterization of the right hand sides $f$ for which the above Dirichlet problem is solvable and has multiple solutions.

We consider the existence of positive solutions of the singular nonlinear semipositone problem of the form $$\left\{\begin{array}{c}-{\mathrm{div}\left(\right|x|}^{-\alpha p}{\left|\nabla u\right|}^{p-2}{\nabla u)=|x|}^{-(\alpha +1)p+\beta }\left(a{u}^{p-1}-f\left(u\right)-{\displaystyle \frac{c}{u^{\gamma }}}\right),x\in \Omega ,\hfill \\ u=0,x\in \partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded smooth domain of ${ℝ}^N$ with $0\in \Omega$, $1<p<N$, $0\le \alpha <(N-p)/p$, $\gamma \in (0,1)$, and $a$, $\beta$, $c$ and $\lambda$ are positive parameters. Here $f:[0,\infty )\to ℝ$ is a continuous function. This model arises in the studies of population biology of one species with $u$ representing the concentration of the species. We discuss the existence of a positive solution when $f$ satisfies certain additional conditions. We use the method of sub-supersolutions...

Let $\Omega$ be a smooth bounded domain in ${ℝ}^N,N\&gt;1$ and let $n\in {\mathbb{N}}^{*}$. We prove here the existence of nonnegative solutions $u_n$ in $BV\left(\Omega \right)$, to the problem $$\left(P_n\right)\left\{\begin{array}{cc}-div\sigma +2n\left({\int }_{\Omega }u-1\right){sign}^{+}\left(u\right)=0\hfill & \text{in}\Omega ,\hfill \\ \sigma ·\nabla u=\left|\nabla u\right|\hfill & \text{in}\Omega ,\hfill \\ u\text{is}\text{not}\text{identically}\text{zero},-\sigma ·\overrightarrow{n}u=u\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\overrightarrow{n}$ denotes the unit outer normal to $\partial \Omega$, and ${\mathrm{sign}}^{+}\left(u\right)$ denotes some $L^{\infty }\left(\Omega \right)$ function defined as: $${\mathrm{sign}}^{+}\left(u\right).u=u^{+},0\le {\mathrm{sign}}^{+}\left(u\right)\le 1.$$ Moreover, we prove the tight convergence of $u_n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-{\Delta }_1$ on $\Omega$ when $n$ goes to $+\infty$.

We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $\Delta ₚu={\left|u\right|}^{p-2}u$ in Ω, ⎨ ⎩ ${\left|\nabla u\right|}^{p-2}\partial u/\partial \nu ={\lambda \varrho \left(x\right)\left|u\right|}^{p-2}u+\mu {\left|u\right|}^{p-2}u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.