Related papers: Unexpected phenomena in a one dimensional elliptic…

We make some remarks on the Euler-Lagrange equation of energy functional $I(u)=\int_\Omega f(\det Du)\,dx,$ where $f\in C^1(\mathbb R).$ For certain weak solutions $u$ we show that the function $f'(\det Du)$ must be a constant over the…

This paper is concerned with the forward and inverse problems for the fractional semilinear elliptic equation $(-\Delta)^s u +a(x,u)=0$ for $0<s<1$. For the forward problem, we proved the problem is well-posed and has a unique solution for…

We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\;…

We prove a number of \textit{a priori} estimates for weak solutions of elliptic equations or systems with vertically independent coefficients in the upper-half space. These estimates are designed towards applications to boundary value…

Let $\lambda^{*}>0$ denote the largest possible value of $\lambda$ such that $$ \{{array}{lllllll} \Delta^{2}u=\frac{\lambda}{(1-u)^{p}} & \{in}\ \ B, 0<u\leq 1 & \{in}\ \ B, u=\frac{\partial u}{\partial n} =0 & \{on}\ \ \partial B. {array}…

In this paper, we investigate the regularity of weak solutions $u\colon\Omega\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $\Omega$}, \end{equation*} whose…

In this article, we study the existence and multiplicity of solutions of the following $(p,q)$-Laplace equation with singular nonlinearity: \begin{equation*} \left\{\begin{array}{rllll} -\Delta_{p}u-\ba\Delta_{q}u & = \la u^{-\de}+ u^{r-1},…

We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \\ u> 0, &x\in…

In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth…

We study Dirichlet problems for fractional Laplace equations of the form $(-\Delta)^{\frac{\alpha}{2}} u = f(x,u)$ in $\mathbb{R}^{n}$ for $0<\alpha<n$ where the nonlinearity $f(x,u) = \sum_{i=1}^{M} \sigma_{i} u^{q_i} + \omega$ involves…

We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,\Omega, \] where $1<p<\infty$, $\Omega$ is an open subset of…

In this paper, we investigate the existence of positive weak solutions to a nonlocal singular elliptic problem under Dirichlet boundary condition. Problem is settled in fractional Musielak-Sobolev spaces with variable order. The main tool…

We consider weak solutions of the adjoint equation for an elliptic operator in nondivergent form, and their asymptotic properties at an interior point. We assume that the coefficients a_{ij} are bounded, measurable, complex-valued functions…

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{equation}\label{00} \left\{\begin{aligned} (-\Delta)^{\frac{1}{2}}u + u &= Q(x)f(u)\;\;\mbox{in}\;\;\R…

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{div}( M(x) \nabla u ) = -\mathrm{div} (E(x) u) + f$ in a bounded domain of $\mathbb R^N$ with $N \ge 3$. We are particularly…

We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} \left\{ \begin{array}{l} D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] u(0)=u(1)=0,…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq1$, let $K$, $M$ be two nonnegative functions and let $\alpha,\gamma>0$. We study existence and nonexistence of positive solutions for singular problems of the form $-\Delta…

We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-\Delta)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1<q<2,\;p>2,\;1<\beta\leq2\;,…

In this work, we study the existence and regularity results of anisotropic elliptic equations with a singular lower order term that grows naturally with respect to the gradient and unbounded coefficients. We take up the following model…

We prove the uniform boundedness of all solutions for a general class of Dirichlet anisotropic elliptic problems of the form $$-\Delta_{\overrightarrow{p}}u+\Phi_0(u,\nabla u)=\Psi(u,\nabla u) +f $$ on a bounded open subset $\Omega\subset…