Related papers: Removability of singularities and superharmonicity…

In this paper we are concerned with positive singular solutions of the following nonlocal Hartree equation $$-\Delta u\!=\Big( \int_{\mathbb{R}^N\setminus \Gamma}\frac{F(u(y))}{|x-y|^\mu}dy \Big)f (u(x)), \quad x\in…

In this paper, we study the existence of a solution for a class of Dirichlet problems with a singularity and a convection term. Precisely, we consider the existence of a positive solution to the Dirichlet problem $$-\Delta_p u =…

In this paper we find a positive weak solution for a semipositone $p(\cdot )$- Laplacian problem. More precisely, we find a solution for the problem \[ \left\{ \begin{array}{cc} -\Delta _{p(\cdot )}u=f(u)-\lambda & \text{in }\Omega \\ u>0 &…

In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-H\'enon equation $$ (-\Delta)^\sigma u = |x|^\alpha u^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin,…

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-\sigma}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range…

In this paper, we are mainly concerned with the physically interesting static Schr\"{o}dinger-Hartree-Maxwell type equations \begin{equation*} (-\Delta)^{s}u(x)=\left(\frac{1}{|x|^{\sigma}}\ast |u|^{p}\right)u^{q}(x)…

We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} + \mu \quad \text{on} \;\; \mathbb{R}^n \] in the…

For $1<p<\infty$, we consider the following problem $$ -\Delta_p u=f(u),\quad u>0\text{ in }\Omega,\quad\partial_\nu u=0\text{ on }\partial\Omega, $$ where $\Omega\subset\mathbb R^N$ is either a ball or an annulus. The nonlinearity $f$ is…

Let $N \geq 3$ and $\Omega \subset \mathbb{R}^N$ be $C^2$ bounded domain. We study the existence of positive solution $u \in H^1(\Omega)$ of \begin{align*} \left\{ \begin{array}{l} -\Delta u + \lambda u = \frac{|u|^{2^*(s)-2}u}{|x-x_1|^s} +…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in…

In this paper, we will give a horizontal gradient estimate of positive solutions of $\Delta_b u = - \lambda u$ on complete noncompact pseudo-Hermitian manifolds. As a consequence, we recapture the Liouville theorem of positive…

In this article, we show the existence of a unique entropy solution to the following problem: \begin{equation} \begin{split} (-\Delta)_{p,\alpha}^su&= f(x)h(u)+g(x) ~\text{in}~\Omega,\\ u&>0~\text{in}~\Omega,\\ u&=…

In this paper, we investigate positive solutions to a class of Laplace equations with a gradient term on a complete, connected, and noncompact Riemannian manifold \((M^n,g)\) with nonnegative Ricci curvature, namely \[-\Delta u =…

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…

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…

We study positive solutions to the fractional semi-linear elliptic equation $$ (- \Delta)^\sigma u = K(x) u^\frac{n + 2 \sigma}{n - 2 \sigma} ~~~~~~ in ~ B_2 \setminus \{ 0 \} $$ with an isolated singularity at the origin, where $K$ is a…

We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-\Delta)^s u=f$ in $\Omega$, $\mathcal N_s u=0$ in $\Omega^c$, then $u$ is $C^\alpha$…

In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-\Delta)_p^s u_\lambda=\lambda u_\lambda^q - b(x)u_\lambda^r \quad \text{in} \;\Omega,\\…

In this paper, we prove a global bifurcation result for the existence of non-radial branches of solutions to the paramterized family of $\Gamma$-symmetric problems $-\Delta u=f(\alpha,z,u)$, $u|_{\partial D}=0$ on the unit disc $D:=\{z\in…

We provide a Hopf boundary lemma for the regional fractional Laplacian $(-\Delta)^s_{\Omega}$, with $\Omega\subset\mathbb{R}^N$ a bounded open set. More precisely, given $u$ a pointwise or weak super-solution of the equation…