Related papers: Classification of isolated singularities of positi…
Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case $q \in (0,1)$ $$\displaystyle \ \ -\Delta u+ u =I_\alpha[u^p] u^q\;\; {\rm in}\; \mathbb{R}^N\setminus\{0\}, %…
The existence of a positive solution to a class of Choquard equations with potential going at a positive limit at infinity possibly from above or oscillating is proved. Our results include the physical case and do not require any symmetry…
Existence results for a class of Choquard equations with potentials are established. The potential has a limit at infinity and it is taken invariant under the action of a closed subgroup of linear isometries of $\mathbb{R}^N$. As a…
This article is concerned with the existence of positive weak solutions for the following quasilinear Schr\"odinger Choquard equation: \begin{equation*} \begin{array}{cc} \displaystyle -div(g^2(u)\nabla u) + g(u)g'(u)\nabla u + a(x) u =…
This paper is devoted to the Q-curvature type equation with singularities; mainly we give existence and regularity results of solutions. To have positive solutions which will be meaningfully in conformal geometry we restrict ourself to…
In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in…
We study solutions to conformally invariant equations with isolated singularties.
In this paper, we give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \quad\quad (-\Delta)^s u = |x|^{\theta} u^{p}\quad {\rm in}\ \ B_1\setminus\{0\},\quad u=…
Using variational methods, we establish the existence of infinitely many solutions to an elliptic problem driven by a Choquard term and a singular nonlinearity. We further show that if the problem has a positive solution, then it is bounded…
In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_{\mu}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu}-2}u, \quad…
In this paper, we study the existence and qualitative properties of positive solutions to a Choquard-type equation with Hardy potential. We develop a nonlocal version of concentration-compactness principle involving the Hardy potential to…
In this paper we consider the existence of positive solutions for a singular elliptic problem involving an asymtotically linear nonlinearity and depending on one positive parameter. Using variational methods, together with comparison…
In this paper, we study the following quasilinear Schr\"{o}dinger equation of Choquard type $$ -\triangle u+V(x)u-\triangle (u^{2})u=(I_\alpha *|u|^p)|u|^{p-2}u, \ \ x \in \mathbb{R}^{N}, $$ where $N\geq 3$,\ $0<\alpha<N$,…
The present paper is concerned with the existence of positive high energy solution of the Choquard equation. Under certain assumptions, the ground state of Choquard equation can not be achieved. However, by global compactness analysis, we…
We compare some algebras appeared in the recent attempts to prove resolution of singularities in positive characteristic. We also construct an algebra which encodes the same information and it is equivalent, up to integral closure, to the…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
In this paper, we consider a class of nonlinear Choquard equations with critical growth, and we show the existence of solutions of the equations. Besides, we consider the corresponding minimizing problem and prove the existence of a…
In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -\Delta u+\frac{\mu}{|x|^2} u=u^p\quad {\rm in}\quad \Omega\setminus\{0\},\qquad…
The paper concerns singular solutions of nonlinear elliptic equations.
We study the local properties of positive solutions of the equation $-\Delta u+ae^{bu}=m|\nabla u|^q$ in a punctured domain $\Omega\setminus\{0\}$ of $\bf R^2$ where $m,a,b$ are positive parameters and $q>1$. We study particularly the…