Related papers: On isolated singularities of Kirchhoff--type Lapla…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

We are concerned with the following Kirchhoff type equation $$-\varepsilon^2 M \left(\varepsilon^{2-N} \int_{\mathbb{R}^N} | \nabla u|^2\, \mathrm{d} x\right) \Delta u+V(x)u = f(u),\ x \in \mathbb{R}^N,\ \ N\ge2, $$ where $M \in…

We generalise and sharpen several recent results in the literature regarding the existence and complete classification of the isolated singularities for a broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm…

This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{equation*} \begin{array}{cc} \displaystyle M\left( \|u\|^2\right) (-\Delta)^s u =…

In this paper, we study the existence of normalized solutions to the following Kirchhoff equation with a perturbation: $$ \left\{ \begin{aligned} &-\left(a+b\int _{\mathbb{R}^{N}}\left | \nabla u \right|^{2} dx\right)\Delta u+\lambda…

This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \begin{equation}\label{ellip}\tag{$\ast$} \Delta u-\beta u+\lambda\frac{e^u}{\int_{\Omega}e^u}=0~\mathrm{in}~\Omega,…

The aim of this paper is to classify the positive solutions of the nonlocal critical equation: $$ -\Delta u=\left(I_{\mu}\ast u^{2^{\ast}_{\mu}}\right)u^{{2}^{\ast}_{\mu}-1},~~x\in\mathbb{R}^{N}, $$ where $0<\mu<N$, if $N=3\ \hbox{or} \ 4$…

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the distance of a point $x\in \Omega$ to the boundary. We study the positive solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$, where $p>0,…

We study the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = \alpha |u|^{p-2}u+\beta |u|^{q-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^N$, with $1<q<p$. We investigate the relation between two critical curves on…

We prove convergence of positive solutions to \[ u_t = u\Delta u + u\int_{\Omega} |\nabla u|^2, \qquad u\rvert_{\partial\Omega} =0, \qquad u(\cdot,0)=u_0 \] in a bounded domain $\Omega\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in…

We study the system $-\Delta \mathbf{u}=| \mathbf{u}|^{\alpha-1} \mathbf{u}$ with $1<\alpha\leq\frac{n+2}{n-2}$, where $ \mathbf{u}=(u_1,\dots,u_m)$, $m\geq 1$, is a $C^2$ nonnegative function that develops an isolated singularity in a…

In this paper we show the uniqueness of the critical point for \emph{semi-stable} solutions of the problem $$\begin{cases} -\Delta u=f(u)&\text{in }\Omega\\ u>0&\text{in }\Omega\\ u=0&\text{on } \partial\Omega,\end{cases}$$ where…

We are concerned with the following Schr\"odinger-Newton problem \begin{equation} -\varepsilon^2\Delta u+V(x)u=\frac{1}{8\pi \varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(\xi)}{|x-\xi|}d\xi\big)u,~x\in \mathbb R^3. \end{equation} For…

In this paper, we consider the existence of positive solutions to the following slightly supercritical Choquard equation \begin{equation*} \begin{cases} -\Delta…

In this work, we investigate the quantitative estimates of the unique continuation property for solutions of an elliptic equation $\Delta u = V u + W_1 \cdot \nabla u + \hbox{div} (W_2 u)$ in an open, connected subset of $\mathbb{R}^d$,…

In the present paper, we apply a global branch approach to study the existence, non-existence and multiplicity of positive normalized solutions $(\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the following Kirchhoff problem $$…

We prove the existence, uniqueness, and sharp bilateral pointwise estimates for positive bounded solutions to the Lane--Emden type problem \[ \begin{cases} L u = \sum\limits_{i=1}^{m}\sigma_{i} u^{q_{i}}+\sigma_0, \quad u\geq0 & \text{in }…

In this paper, we consider the following singularly perturbed fractional Kirchhoff problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N}…

We construct positive singular solutions for the problem $-\Delta u=\lambda \exp (e^u)$ in $B_1\subset \mathbb{R}^n$ ($n\geq 3$), $u=0$ on $\partial B_1$, having a prescribed behaviour around the origin. Our study extends the one in Y.…

In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -\Delta u= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~\Omega},\\[1mm] u>0,~ &{\text{in}~\Omega},\\[1mm] u=0, &{\text{on}~\partial…