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Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ \tau v_t = \Delta v - v + u \end{cases} \end{align*} in a ball $\Omega \subset \mathbb R^n$, $n…

In this paper we consider semilinear equations $-\Delta u=f(u)$ with Dirichlet boundary conditions on certain convex domains of the two dimensional model spaces of constant curvature. We prove that a positive, semi-stable solution $u$ has…

In this work we consider an energy subcritical semi-linear wave equation ($3 < p < 5$) \[ \partial_t^2 u - \Delta u = \phi(x) |u|^{p-1} u, \qquad (x,t) \in {\mathbb R}^3 \times {\mathbb R} \] with initial data $(u,u_t)|_{t=0} = (u_0,u_1)\in…

We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the…

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…

Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), &…

In this paper we establish a new existence result for the quasilinear elliptic problem \[ -{\rm div}(A(x,u)|\nabla u|^{p-2}\nabla u) +\frac1p A_t(x,u)|\nabla u|^p + V(x)|u|^{p-2} u = g(x,u)\quad\mbox{ in } \mathbb{R}^N, \] with $N\ge 2$,…

We are interested in the following Dirichlet problem $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\mathrm{dist}\,(x,\mathbb{R}^N \setminus \Omega)^2} = f(x,u) & \quad \mbox{in } \Omega \\ u = 0 &…

In this paper, we investigate interior gradient estimates for solutions to the mean curvature equation $$ \dive \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = f(\nabla u)$$ under various nonlinear assumptions on the right-hand…

This work deals with the existence of at least two positive solutions for the class of quasilinear elliptic equations with cylindrical singularities and multiple critical nonlinearities that can be written in the form \begin{align*}…

In this article we prove that the first eigenvalue of the $\infty-$Laplacian $$ \left\{ \begin{array}{rclcl} \min\{ -\Delta_\infty v,\, |\nabla v|-\lambda_{1, \infty}(\Omega) v \} & = & 0 & \text{in} & \Omega v & = & 0 & \text{on} &…

We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case…

In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{(1-u)^\gamma}=g & \mbox{in $\Omega$,}\newline \hfill u=0 \hfill & \mbox{on $\partial\Omega$,}…

It is shown that for each finite number of Dirac measures supported at points $s_n$ in three-dimensional Euclidean space, with given amplitudes $a_n$, there exists a unique real-valued Lipschitz function $u$, vanishing at infinity, which…

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) =…

This article deals with the study of the following Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left(\, \int\limits_{\mathbb{R}^N}|\nabla u|^p\right) (-\Delta_p) u + V(x)|u|^{p-2}u = \left(\,…

In this paper, the semilinear elliptic systems with Dirichlet boundary value are considered \begin{align} \left\{\begin{array}{ll} -\Delta v=f(u) & \mathrm{in}\ \Omega, -\Delta u=g(v) & \mathrm{in}\ \Omega, u=0, \ v=0 & \mathrm{on}\…

In this work, first we prove that for any compact set $K\subset\mathbb{R}^{n}$ and any continuous function $\phi$ defined on $\partial K$, there exists a bounded weak solution in $C(\bar{\mathbb{R}^{n}\backslash K}) \cap…

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0…

We construct weak solutions to the Navier-Stokes inequality, $$ u\cdot \left(\partial_t u -\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \leq 0 $$ in $\mathbb{R}^3$, which blow up at a single point $(x_0,T_0)$ or on a set $S \times…