Related papers: Pointwise estimates of Brezis-Kamin type for solut…

We give necessary and sufficient conditions for the existence of weak solutions to the model equation $$-\Delta_p u=\sigma \, u^q \quad \text{on} \, \, \, \R^n,$$ in the case $0<q<p-1$, where $\sigma\ge 0$ is an arbitrary locally integrable…

We study quasilinear elliptic equations of the type $-\Delta_{p} u = \sigma u^{q} + \mu \; \; \text{in} \;\; \bf{R}^n$ in the case $0<q< p-1$, where $\mu$ and $\sigma$ are nonnegative measurable functions, or locally finite measures, and…

We discuss recent advances in the theory of quasilinear equations of the type $ -\Delta_{p} u = \sigma u^{q} \; \; \text{in} \;\; \mathbb{R}^n, $ in the case $0<q< p-1$, where $\sigma$ is a nonnegative measurable function, or measure, for…

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega,\] for $0 < q < 1$, where $\sigma\ge 0$ is a measurable function, or a Radon…

We study finite energy solutions to quasilinear elliptic equations of the type $$ -\Delta_pu=\sigma \, u^q \quad \text{in } \mathbb{R}^n,$$ where $\Delta_p$ is the $p$-Laplacian, $p>1$, and $\sigma$ is a nonnegative function (or measure) on…

We give necessary and sufficient conditions for the existence of a BMO solution to the quasilinear equation $-\Delta_{p} u = \mu$ in $\mathbb{R}^n$, $u\ge 0$, where $\mu$ is a locally finite Radon measure, and $\Delta_{p}u=…

We give a survey of nonlinear potential estimates and their applications obtained recently for positive solutions to sublinear problems of the type \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega, \] where $0 < q < 1$,…

We study the existence of positive solutions to quasilinear elliptic equations of the type \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{in} \ \mathbb{R}^{n}, \] in the sub-natural growth case $0 < q < p - 1$, where $\Delta_{p}u =…

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…

We investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right…

Local and global pointwise gradient estimates are obtained for solutions to the quasilinear elliptic equation with measure data $-\operatorname{div}(A(x,\nabla u))=\mu$ in a bounded and possibly nonsmooth domain $\Omega$ in $\mathbb{R}^n$.…

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth…

We present a necessary and sufficient condition on nonnegative Radon measures $\mu$ and $\nu$ for the existence of a positive continuous solution of the Dirichlet problem for the sublinear elliptic equation $-\Delta u=\mu u^q+\nu$ with…

For $ p \in (1,N)$ and a domain $\Omega$ in $\mathbb{R}^N$, we study the following quasi-linear problem involving the critical growth: \begin{eqnarray*} -\Delta_p u - \mu g|u|^{p-2}u = |u|^{p^{*}-2}u \ \mbox{ in } \mathcal{D}_p(\Omega),…

In this paper the Nash-Moser iteration method is used to study the gradient estimates of solutions to the quasilinear elliptic equation $\Delta_p u-|\nabla u|^q+b(x)|u|^{r-1}u=0$ defined on a complete Riemannian manifold $(M,g)$. When…

We introduce a class of weak solutions to the quasilinear equation $-\Delta_p u = \sigma |u|^{p-2}u$ in an open set $\Omega\subset\mathbf{R}^n$. Here $p>1$, and $\Delta_p u$ is the $p$-Laplacian operator. Our notion of solution is tailored…

We extend the so-called universal potential estimates of the Kuusi-Mingione type (J.Funct. Anal. 2012) to the singular case $1<p\leq 2-1/n$ for the quasilinear equation with measure data \begin{equation*} -\operatorname{div}(A(x,\nabla…

In this article we study local and global properties of positive solutions of $-\Delta_mu=|u|^{p-1}u+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, with $m>1$, $p,q>0$ and $M\in\mathbb R$. Following some ideas used in…

We characterize the existence of solutions to the quasilinear Riccati type equation \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + \sigma \quad \text{in} ~\Omega, \\ u&=&0 \quad…

In this article, we study the existence of positive solutions to elliptic equation (E1) $$(-\Delta)^\alpha u=g(u)+\sigma\nu \quad{\rm in}\quad \Omega,$$ subject to the condition (E2) $$u=\varrho\mu\quad {\rm on}\quad \partial\Omega\ \ {\rm…