Related papers: Good-$\lambda$ type bounds of quasilinear elliptic…

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of $p$-$q$ type and singular nonlinearities \begin{equation*} \left\{…

In this manuscript we study geometric regularity estimates for quasi-linear elliptic equations of $p$-Laplace type ($1 < p< \infty$) with strong absorption condition: $$ -\text{div}\,(\Phi(x, u, \nabla u)) + \lambda_0(x) u_{+}^q(x) = 0…

In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -\Delta_{p_1}u_1 & =-|u_1|^{p_1-2}u_1 \quad && \text{in } \Omega,\newline -\Delta_{p_2}u_2 & =-|u_2|^{p_2-2}u_2 \quad && \text{in }…

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N,…

We study the existence of solutions of the nonlinear problem $$ \left\{ \begin{alignedat}{2} -\Delta u + g(u) & = \mu & & \quad \text{in } \Omega,\\ u & = 0 & & \quad \text{on } \partial \Omega, \end{alignedat} \right. $$ where $\mu$ is a…

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\…

In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain…

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is \begin{equation*} \begin{cases} -\Delta_{p} u -\text{div} (c(x)|u|^{p-2}u)) =f & \text{in}\ \Omega, \\ \left( |\nabla…

In the present paper we investigate the following semilinear singular elliptic problem: \begin{equation*} (\rm P)\qquad \left \{\begin{array}{l} -\Delta u = \dfrac{p(x)}{u^{\alpha}}\quad \text{in} \Omega \\ u = 0\ \text{on} \Omega,\ u>0…

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 $N\ge 2$,…

Let $\Omega$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, \lambda>0$ and $0<\delta<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $\chi_A$. We…

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is $$\begin{cases} -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega, u=0 &\text{on}\ \partial\Omega, u>0…

We consider the problem $$ (P_\lambda)\quad -\Delta_{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }\Omega $$ under Dirichlet or Neumann boundary conditions. Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$…

We study the existence/nonexistence of positive solution to the problem of the type: \begin{equation}\tag{$P_{\lambda}$} \begin{cases} \Delta^2u-\mu a(x)u=f(u)+\lambda b(x)\quad\textrm{in $\Omega$,}\\ u>0 \quad\textrm{in $\Omega$,}\\…

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 consider model semilinear elliptic equations of the type \[ \begin{cases} - \mathrm{div} (A(x) \nabla u) = f u^{- \lambda}, \quad u > 0 \quad \text{in} \ \Omega, \\ u \in H_{0}^{1}(\Omega), \end{cases} \] where $\Omega$ is a bounded…

We study quasilinear degenerate singular elliptic equation of type -Delta_p u = \frac{u^{p^*(s)-1}}{|y|^t}$ in a smooth bounded domain \Omega in R^n=R^k \times R^{N-k}$, x=(y,z) in R^k \times R^{N-k}, 2 \leq k<N and N \geq 3, 1<p<2, 0\leq…

Let (M,g) be a smooth connected compact Riemannian manifold of finite dimension n \geq 2 with a smooth boundary \partial M. We consider the problem -{\epsilon}^2\Delta_gu+u=|u|^{p-2}u, u>0 on M, \partial u/ \partial{\nu}=0 on \partial M…

We study positive solutions of equation (E1) $-\Delta u + u^p|\nabla u|^q= 0$ ($0\leq p$, $0\leq q\leq 2$, $p+q>1$) and (E2) $-\Delta u + u^p + |\nabla u|^q =0$ ($p>1$, $1<q\leq 2$) in a smooth bounded domain $\Omega \subset \mathbb{R}^N$.…

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 $\mathbb{R}^n$.…