We study stationary solutions of the system $u_t=\nabla ((m-1)/m\nabla u^m+u\nabla \phi )$, m => 1, Δφ = ±u, defined in a bounded domain Ω of ${ℝ}^n$. The physical interpretation of the above system comes from the porous medium theory and semiconductor physics.

We study properties of Lipschitz truncations of Sobolev functions with constant and variable exponent. As non-trivial applications we use the Lipschitz truncations to provide a simplified proof of an existence result for incompressible power-law like fluids presented in [Frehse et al., SIAM J. Math. Anal34 (2003) 1064–1083]. We also establish new existence results to a class of incompressible electro-rheological fluids.

We study the existence of nonnegative solutions of elliptic equations involving concave and critical Sobolev nonlinearities. Applying various variational principles we obtain the existence of at least two nonnegative solutions.

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems $$\left\}\begin{array}{cccc}-{\Delta }_{p}u=f(x,u,v),\hfill & \text{in}\Omega ,-{\Delta }_{p}v=g(x,u,v),\hfill & \text{in}\Omega ,u=v=0,& \text{on}\partial \Omega ,\end{array}\right.$$ where $-{\Delta }_p$ is the $p$-Laplace operator, $p>1$ and $\Omega$ is a $C^{1,\alpha }$-domain in ${ℝ}^n$. We prove an analogue of [7, 16] for the eigenvalue problem with $f(x,u,v)={\lambda }_1{v}^{p-1}$, $g(x,u,v)={\lambda }_2{u}^{p-1}$ and obtain a non-existence result of positive solutions for the general systems.

Let Ω be a smooth bounded domain in ${𝐑}^n$, n > 1, let a and f be continuous functions on $\overline{\Omega }$, $1^{☆}=\frac{n}{n-1}$. We are concerned here with the existence of solution in $BV\left(\Omega \right)$, positive or not, to the problem:
$$\left\{\begin{array}{cccc}\hfill -\mathrm{div}\sigma +a\left(x\right)signu& amp;={f\left|u\right|}^{1^{☆}-2}u\sigma .\nabla u\hfill & amp;=\left|\nabla u\right|\mathrm{in}\Omega u\mathrm{is}\mathrm{not}\mathrm{identically}\mathrm{zero},& amp;-\sigma .n\left(u\right)=\left|u\right|\mathrm{on}\partial \Omega .\end{array}\right.$$ This problem is closely related to the extremal functions for the problem of the best constant of $W^{1,1}\left(\Omega \right)$ into $L^{\frac{N}{N-1}}\left(\Omega \right)$.

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition$${\int }_1^{\infty }\frac{P\left(t\right)}{t^2}\mathrm{d}t=\infty .$$∫ 1 ∞ P ( t ) t 2   d t = ∞ .

We study the regularity of finite energy solutions to degenerate n-harmonic equations. The function K(x), which measures the degeneracy, is assumed to be subexponentially integrable, i.e. it verifies the condition exp(P(K)) ∈ Lloc1. The function P(t) is increasing on  [0,∞[  and satisfies the divergence condition ${\mathrm{\int }}_{\mathrm{1}}^{\mathrm{\infty }}\frac{\mathit{P}\mathrm{\left(}\mathit{t}\mathrm{\right)}}{{\mathit{t}}^{\mathrm{2}}}\hspace{0.17em}\mathrm{d}\mathit{t}\mathrm{=}\mathrm{\infty }\mathit{.}$

Let $\Omega \subset {ℝ}^n,n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic measure...