Related papers: Normalized solutions for critical Choquard systems

This paper is devoted to studying the following nonlinear biharmonic Schr\"odinger equation with combined power-type nonlinearities \begin{equation*} \begin{aligned} \Delta^{2}u-\lambda u=\mu|u|^{q-2}u+|u|^{4^*-2}u\quad\mathrm{in}\…

In this paper, we study the existence of ground state solutions to the following p-Laplacian equation in some dimension $N\geq3$ with an $L^2$ constraint: \begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u \quad…

In this paper, we study the following fractional Choquard system \begin{align*} \begin{split} \left\{ \begin{array}{ll} (-\Delta)^{1/2}u=\lambda_1 u+(I_\mu*F(u,v))F_u (u,v), \quad\mbox{in}\ \ \mathbb{R}, (-\Delta)^{1/2}v=\lambda_2…

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schr\"{o}dinger equation with general nonlinearity: Mass super-critical case: \[\begin{cases} -\Delta u+V(x)u+\lambda…

In this paper, we study the existence of normalized solutions for the following quasilinear Schr\"odinger equation with Sobolev critical exponent: \begin{eqnarray*} -\Delta u-u\Delta (u^2)+\lambda…

In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…

In this paper we study the following fractional Choquard equation with mixed nonlinearities: \[ \left\{ \begin{array}{l} (-\Delta)^s u = \lambda u + \alpha \left( I_\mu * |u|^q \right) |u|^{q-2} u + \left( I_\mu * |u|^p \right) |u|^{p-2} u,…

This paper is concerned with the quantitative stability of critical points of the Hardy-Littlewood-Sobolev inequality. Namely, we give quantitative estimates for the Choquard equation: $$-\Delta u=(I_{\mu}\ast|u|^{2_\mu^*}) u^{2_\mu^*-1}\ \…

In the present paper, we study the existence of normalized solutions to the following Kirchhoff type equations \begin{equation*} -\left(a+b\int_{\R^3}|\nabla u|^2\right)\Delta u+V(x)u+\lambda u=g(u)~\hbox{in}~\R^3 \end{equation*} satisfying…

The paper is concerned with the existence and asymptotic properties of normalized ground states of the following nonlinear Schr\"odinger system with critical exponent: \begin{equation*} \left\{\begin{aligned} &-\delta u+\lambda_1…

We investigate normalized solutions with prescribed $L^2$-norm for the upper critical fractional Choquard equation \[(-\Delta)^s u+V(\varepsilon x)u=\lambda…

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation \[ -{\Delta}u+ u=\big(I_\alpha*|u|^{\frac{\alpha}{N}+1}\big)|u|^{\frac{\alpha}{N}-1}u+f(x,u)\qquad…

In this paper, we prove the existence of positive solutions $(\lambda_1,\lambda_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schr\"odinger system $$\begin{cases} -\Delta u + \lambda_1 u= \mu_1|u|^{p-2}u+\beta v \quad…

Given $\mu>0$ we look for solutions $ \lambda\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -\Delta v_i+ \lambda v_i+V_i(x)v_i = \sum_{\substack{j=1}}^k\beta_{ij} v_iv_j^2 &\text{ in }…

In present paper, we prove the existence of solutions $(\lambda, u)\in \R\times H^1(\R^N)$ to the following Schr\"odinger equation $$ \begin{cases} -\Delta u(x)+V(x)u(x)+\lambda u(x)=g(u(x))\quad &\hbox{in}~\R^N\\ 0\leq u(x)\in H^1(\R^N),…

In this paper, we study the following biharmonic Choquard type equation \begin{align*} \begin{split} \left\{ \begin{array}{ll} \gamma\Delta^2u-\beta\Delta u=\lambda u+(I_\mu*F(u))f(u), \quad\mbox{in}\ \ \mathbb{R}^4,…

In this paper, we consider the following linearly coupled Kirchhoff--Choquard system in $\mathbb{R}^3$: \begin{align*} \begin{cases} -\left(a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2\,dx\right)\Delta u + V_1 u = \mu (I_{\alpha} * |u|^p)…

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem \begin{cases} -\Delta u + \lambda u = g(u) \quad \mathrm{in} \ \mathbb{R}^N, \ N \geq 3, \\ u \in H^1(\mathbb{R}^N), \\…

In this article, we explore the fractional Kirchhoff-Choquard system given by $$ \left\{ \begin{array}{lr} (a+b\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}} u|^2\;dx)(-\Delta)^su=\lambda_1u+(I_{\mu}*|v|^{{2^*_{\mu,s}}})|u|^{{2^*_{\mu,s}}-2}u…

In this paper we are interested in the following nonlinear Choquard equation $$ -\Delta u+(\lambda V(x)-\beta)u =\big(|x|^{-\mu}\ast |u|^{2_{\mu}^{\ast}}\big)|u|^{2_{\mu}^{\ast}-2}u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^N, $$…