In this article, we study the following anisotropic p-Laplacian equation with variable exponent given by \begin{equation*} (P) \begin{cases} -\Delta_{H,p}u =\frac{\lambda f(x)}{u^{q(x)}}+g(u) \ \text{ in }\Omega,\\ u > 0 }\Omega,\ u=0\text{ on }\partial\Omega, \end{cases} \end{equation*} under assumption $\Omega$ is a bounded smooth domain $\mathbb{R}^N$ $p,N\geq 2$, $\lambda>0$ and $0 < q \in ...

Searching minimizers of functions on the convenient level set of the constrain function we obtain weak solutions of a non-homogenuous p-Laplace equation in border case without using the regularity results of linear elliptic equations .

Consider two perfectly conducting spheres in a homogeneous medium where the current-electric field relation is the power law. Electric field E blows up in the L∞-norm as δ, the distance between the conductors, tends to zero. We give here a concise rigorous justification of the rate of this blow-up in terms of δ. If the current-electric field relation is linear, see similar results obtained earl...

We study the behavior of solutions to a Schrödinger equation with large, rapidly oscillating, mean zero, random potential with Gaussian distribution. We show that in high dimension d > m, where m is the order of the spatial pseudo-differential operator in the Schrödinger equation (with m = 2 for the standard Laplace operator), the solution converges in the L sense uniformly in time over finite ...