Related papers: Doubling the equatorial for the prescribed scalar …

In this paper, we consider the following Schr\"{o}dinger equation: \begin{equation*} \begin{cases} -\Delta u=\lambda u+\mu|u|^{q-2}u+|u|^{2^*-2}u\quad\text{in }\mathbb{R}^N,\\ \int_{\mathbb{R}^N}|u(x)|^2dx=a,\quad u\in H^1(\mathbb{R}^N),\\…

In this paper we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{equation*} \begin{array}{rl} (-\Delta)^s u &= \displaystyle-\lambda|u|^{q-2}u +…

In this work, we establish the multiplicity of positive solutions for the following critical fractional Choquard equation with a perturbation on the star-shaped bounded domain $$ \left\{ \begin{array}{lr} (-\Delta)^s u = \lambda u…

We study the existence of solutions to the problem $$ (-\Delta)^{\frac{n}{2}}u = Qe^{nu}\quad\text{in }\mathbb{R}^n, \quad V := \int_{\mathbb{R}^n}e^{nu}dx < \infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and…

We study the existence of radially symmetric solutions of the following nonlinear scalar field equations in $\mathbb R^N$ ($N \geq 2$): $$ (*)_m \quad - \Delta u + \mu u = g(u) \quad \text{in}\ {\mathbb R}^N, \quad {1\over 2} \int_{{\mathbb…

We consider the multi-bump solutions of the following fractional Nirenberg problem \begin{equation}\label{01} (-\Delta)^s u=K(x)u^{\frac{n+2s}{n-2s}}, \;\;\;\;u>0\;\;\text{ in }\mathbb{R}^n, \end{equation} where $s\in (0,1)$ and $n>2+2s$.…

We study the following fractional Schr\"{o}dinger equation \begin{equation*}\label{eq0.1} \epsilon^{2s}(-\Delta)^s u + V(x)u = |u|^{p - 2}u, \,\,x\in\,\,\mathbb{R}^N, \end{equation*} where $s\in (0,\,1)$, $N>2s$, $p>1$ is subcritical and…

We consider a nonautonomous semilinear elliptic problem where the power nonlinearity is multiplied by a discontinuous coefficient that equals one inside a bounded open set $\Omega$ and it equals minus one in its complement. In the slightly…

In the present work, we establish the existence of two positive solutions for singular nonlocal elliptic systems. More precisely, we consider the following nonlocal elliptic problem: $$\left\{\begin{array}{lll} (-\Delta)^su +V_1(x)u =…

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.…

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} +…

We construct unbounded positive $C^2$-solutions of the equation $\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in ${\R}^n$ (equipped with Euclidean metric $g_o$) such that $K$ is bounded between two positive numbers in ${\R}^n$, the conformal…

In this work we study the nonnegative solutions of the elliptic system \Delta u=|x|^{a}v^{\delta}, \Delta v=|x|^{b}u^{\mu} in the superlinear case \mu \delta>1, which blow up near the boundary of a domain of R^{N}, or at one isolated point.…

This paper is concerned with the following fractional Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{s} u+u= k(x)f(u)+h(x) \mbox{ in } \mathbb{R}^{N}\\ u\in H^{s}(\R^{N}), \, u>0 \mbox{ in } \mathbb{R}^{N},…

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…

It is established existence and multiplicity of solution for the following class of quasilinear elliptic problems $$ \left\{ \begin{array}{lr} -\Delta_\Phi u = \lambda a(x) |u|^{q-2}u + |u|^{p-2}u, & x\in\Omega, u = 0, & x \in \partial…

We build infinitely many geometrically distinct non-radial sign-changing solutions for the Hamiltonian-type elliptic systems $$ -\Delta u =|v|^{p-1}v\ \hbox{in}\ \mathbb{R}^N,\ -\Delta v =|u|^{q-1}u\ \hbox{in}\ \mathbb{R}^N,$$ where the…

This paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we…

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as…

In this article, we show that the solution to defocusing cubic nonlinear Schr\"odinger equation (NLS) posed on the two-dimensional waveguide \begin{align*} i\partial_tu+\Delta_{\R\times\T}u=|u|^2u \end{align*} is globally well-posed in…