Related papers: Normalized solutions of linearly coupled Choquard …

In this paper, we study the existence of normalized solutions to the following nonlinear Schr\"{o}dinger systems with exponential growth \begin{align*} \left\{ \begin{aligned} &-\Delta u+\lambda_{1}u=H_{u}(u,v), \quad \quad \hbox{in…

We investigate the following problem $$ -{\rm div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-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 elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 +…

In this paper, we are concerned with the existence and asymptotic behavior of least energy solutions for following nonlinear Choquard equation driven by fractional Laplacian $$(-\Delta)^{s} u+\lambda V(x)u=(I_{\alpha}\ast F(u))f(u) \ \ in \…

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$…

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schr\"{o}dinger equation $$ \left\{ \begin{array}{lc} -\Delta u+ \lambda V(x)u=u \log u^2, & \mbox{in} \quad \mathbb{R}^{N}, \\ u \in…

We prove the existence of solutions $(\lambda, v)\in \mathbb{R}\times H^{1}(\Omega)$ of the elliptic problem \[ \begin{cases} -\Delta v+(V(x)+\lambda) v =v^{p}\ &\text{ in $ \Omega, $} \ v>0,\qquad \int_\Omega v^2\,dx =\rho. \end{cases} \]…

We study the existence of positive solutions for the following class of $(p,q)$-Laplacian coupled systems \[ \left\{ \begin{array}{lr} -\Delta_{p} u+a(x)|u|^{p-2}u=f(u)+ \alpha\lambda(x)|u|^{\alpha-2}u|v|^{\beta}, & x\in\mathbb{R}^{N},…

In this paper we study the following nonlinear Choquard equation $$ -\Delta u+u=\left(\ln\frac{1}{|x|}\ast F(u)\right)f(u),\quad\text{ in }\,\mathbb{R}^2, $$ where $f\in C^1(\mathbb{R})$ and $F$ is the primitive of the nonlinearity $f$…

In this work we study the following class of systems of coupled nonlinear fractional nonlinear Schr\"odinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1+ \lambda_1 u_1= \mu_1 |u_1|^{2p-2}u_1+\beta |u_2|^{p}…

We find a normalized solution $u=(u_1,\ldots,u_K)$ to the system of $K$ coupled nonlinear Schr\"odinger equations \begin{equation*} \left\{ \begin{array}{l} -\Delta u_i+ \lambda_i u_i = \sum_{j=1}^K\beta_{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2}…

In the present paper, we study the normalized solutions with least energy to the following system: $$\begin{cases} -\Delta u+\lambda_1u=\mu_1 |u|^{p-2}u+\beta r_1|u|^{r_1-2}|v|^{r_2}u\quad &\hbox{in}\;\mathbb R^N,\\ -\Delta…

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$ -\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2, $$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the…

We study a weakly coupled supercritical elliptic system of the form \begin{equation*} \begin{cases} -\Delta u = |x_2|^\gamma \left(\mu_{1}|u|^{p-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u \right) & \text{in }\Omega,\\ -\Delta v =…

We establish the existence of positive normalized (in the $L^2$ sense) solutions to non-variational weakly coupled elliptic systems of $\ell$ equations. We consider couplings of both cooperative and competitive type. We show the problem can…

This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation \begin{equation} \tag{$P$} \label{equ1} -\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N), \end{equation} and its…

This article establishes the existence and multiplicity of normalized solutions to the weighted nonlinear Schr\"odinger-type equation governed by the Caffarelli-Kohn-Nirenberg operator, $$ -\text{div}(|x|^{-2a}\nabla u)=\lambda…

In this work, we shall investigate existence and multiplicity of solutions for a nonlocal elliptic systems driven by the fractional Laplacian. Specifically, we establish the existence of two positive solutions for following class of…

In this paper, we study the following fractional Schr\"{o}dinger equation with prescribed mass \begin{equation*} \left\{ \begin{aligned} &(-\Delta)^{s}u=\lambda u+a(x)|u|^{p-2}u,\quad\text{in $\mathbb{R}^{N}$},\\…