Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in ${ℝ}^N$.

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.

Our main purpose is to establish the existence of weak solutions of second order quasilinear elliptic systems ⎧ $-{\Delta }_{p}u+{\left|u\right|}^{p-2}u=f_{1\lambda ₁}{\left(x\right)\left|u\right|}^{q-2}u+2\alpha /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha -2}u{\left|v\right|}^{\beta }$, x ∈ Ω, ⎨ $-{\Delta }_{p}v+{\left|v\right|}^{p-2}v=f_{2\lambda ₂}{\left(x\right)\left|v\right|}^{q-2}v+2\beta /(\alpha +\beta )g_{\mu }{\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v$, x ∈ Ω, ⎩ u = v = 0, x∈ ∂Ω, where 1 < q < p < N and $\Omega \subset {ℝ}^N$ is an open bounded smooth domain. Here λ₁, λ₂, μ ≥ 0 and $f_{i{\lambda }_i}\left(x\right)={\lambda }_i{f}_{i+}\left(x\right)+f_{i-}\left(x\right)$ (i = 1,2) are sign-changing functions, where $f_{i\pm }\left(x\right)=max\pm f_i\left(x\right),0$, $g_{\mu }\left(x\right)=a\left(x\right)+\mu b\left(x\right)$, and ${\Delta }_{p}u={div\left(\right|\nabla u|}^{p-2}\nabla u)$ denotes the p-Laplace operator. We use variational methods.

In this work, by using the Mountain Pass Theorem, we give a result on the existence of solutions concerning a class of nonlocal $p$-Laplacian Dirichlet problems with a critical nonlinearity and small perturbation.

Let $\Omega \subset {ℝ}^n$ be a bounded starshaped domain and consider the $(p,q)$-Laplacian problem $$\begin{array}{c}\hfill -{\Delta }_{p}u-{\Delta }_{q}u=\lambda \left(𝐱\right){\left|u\right|}^{p^{☆}-2}u+\mu {\left|u\right|}^{r-2}u\end{array}$$ where $\mu$ is a positive parameter, $1<q\le p<n$, $r\ge p^{☆}$ and $p^{☆}:=\frac{np}{n-p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p,q)$-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

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 $+\infty$.