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Let $\Omega\subset\mathbb R^{n}\ (n\geq2)$ be either an open ball $B_R$ centred at the origin or the whole space. We study the existence of positive, radial solutions of quasilinear elliptic systems of the form \begin{equation*} \left\{…

This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta u-\lambda \frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=\kappa\alpha…

In the present paper, we consider the coupled Schr\"{o}dinger systems with critical exponent: \begin{equation*} \begin{cases} -\Delta u_i+\lambda_{i}u_i=\sum\limits_{j=1}^{d} \beta_{ij}|u_j|^{3}|u_i|u_i \quad ~\text{ in } \Omega,\\ u_i \in…

Given a smooth bounded domain $\Omega\subset \mathbb R^3$, we consider the following nonlinear Schr\"odinger-Poisson type system \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+ \phi u -\abs{u}^{p-2}u = \omega u & \quad \text{in }…

In this paper, we consider the following Kirchhoff type problem $$\left\{\aligned&-\biggl(a + b\int_{\mathbb{R}^N} |\nabla u|^2 dx \biggr) \Delta u + V(x) u = |u|^{p-2}u &\text{ in } \mathbb{R}^N,\cr &u\in H^1(\mathbb{R}^N),…

In this paper, our aim is to prove the existence of normalized ground state for the following Schr\"odinger systems with potentials $$\begin{cases} -\Delta u_1+V_1(x)u_1+\lambda_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\…

This paper treats the existence of positive solutions of $-\Delta u + V(x) u = \lambda f(u)$ in $\mathbb{R}^N$. Here $N \geq 1$, $\lambda > 0$ is a parameter and $f(u)$ satisfies conditions only in a neighborhood of $u=0$. We shall show the…

We are concerned with the following nonlinear Schr\"odinger equation \begin{eqnarray*} \begin{aligned} \begin{cases} -\Delta u+\lambda u=f(u) \ \ {\rm in}\ \mathbb{R}^{2},\\ u\in H^{1}(\mathbb{R}^{2}),~~~ \int_{\mathbb{R}^2}u^2dx=\rho,…

We are concerned with solutions of the following quasilinear Schr\"odinger equations \begin{eqnarray*} -{\mathrm{div}}\left(\varphi^{2}(u) \nabla u\right)+\varphi(u) \varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u), \quad x \in…

We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…

In this note, we study symmetry of solutions of the elliptic equation \begin{equation*} -\Delta _{\mathbb{S}^{2}}u+3=e^{2u}\ \ \hbox{on}\ \ \mathbb{S}^{2}, \end{equation*} that arises in the study of rigidity problem of Hawking mass in…

In this paper, we give a complete study on the existence and non-existence of normalized solutions for Schr\"{o}dinger system with quadratic and cubic interactions. In the one dimension case, the energy functional is bounded from below on…

We consider the elliptic system $$-\Delta u_i = u_i^3+\sum\limits_{j=1\atop j\not=i}^{q+1}{ \beta_{ij}}u_i u_j^2\ \hbox{in}\ \mathbb R^4, \ i=1,\dots,q+1.$$ when $\alpha:=\beta_{ij}$ and $\beta:=\beta_{i(q+1)}=\beta_{(q+1)j}$ for any…

We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: \[\left\{\begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u + \psi u = \lambda |u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill -…

Orbital stability property for weakly coupled nonlinear Schr\"odinger equations is investigated. Different families of orbitally stable standing waves solutions will be found, generated by different classes of solutions of the associated…

In this paper we deal with the nonlinear Schr\"odinger system \[ -\Delta u_i =\mu_i u_i^3 + \beta u_i \sum_{j\neq i} u_j^2 + \lambda_i u_i, \qquad u_1,\ldots, u_m\in H^1_0(\Omega) \] in dimension 4, a problem with critical Sobolev exponent.…

In this paper we prove the existence of positive solutions of the following singular quasilinear Schr\"{o}dinger equations at critical growth \begin{eqnarray*} -\Delta u-\lambda c(x)u-\kappa\alpha(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u =…

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$,…

Given $\rho>0$, we study the elliptic problem \[ \text{find } (U,\lambda)\in H^1_0(\Omega)\times \mathbb{R} \text{ such that } \begin{cases} -\Delta U+\lambda U=|U|^{p-1}U \int_{\Omega} U^2\, dx=\rho, \end{cases} \] where…

This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schr\"odinger equation \begin{equation*} \begin{aligned} -\Delta u-u\Delta u^2 +\lambda u=|u|^{p-2}u \quad\mathrm{in}\ \mathbb{R}^{N},…