相关论文: Symmetry and classification of positive standing w…

We consider systems of the form \[ \left\{ \begin{array}{l} -\Delta u + u = \frac{2p}{p+q}(I_\alpha \ast |v|^{q})|u|^{p-2}u \ \ \textrm{ in } \mathbb{R}^N, \\ -\Delta v + v = \frac{2q}{p+q}(I_\alpha \ast |u|^{p})|v|^{q-2}v \ \ \textrm{ in }…

In this paper, we study the following coupled Choquard system in $\mathbb R^N$: $$\left\{\begin{align}&-\Delta u+A(x)u=\frac{2p}{p+q} \bigl(I_\alpha\ast |v|^q\bigr)|u|^{p-2}u,\\ &-\Delta…

In this paper, we investigate the existence of positive solution for the following class of elliptic equation $$ - \epsilon^{2}\Delta u +V(x)u= f(u) \,\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where $\epsilon >0$ is a positive parameter,…

We study existence and stability of standing waves for coupled nonlinear Hartree type equations \[ -i\frac{\partial}{\partial t}\psi_j=\Delta \psi_j+\sum_{k=1}^m \left(W\star |\psi_k|^p \right)|\psi_j|^{p-2}\psi_j, \] where…

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},…

The paper deals with the equation $-\Delta u+a(x) u +b(x)u^q -u^p = 0$, $u \in H^1(\R^N)$, whith $N\ge 2$, $1<q<p,\ p<{N+2\over N-2}$ if $N\ge 3$, $\inf a>0$, $a(x)\to a_\infty$ and $b(x)\to 0$ as $|x|\to\infty$. When $a(x)\le a_\infty$ and…

This paper considers a pair of coupled nonlinear Helmholtz equations \begin{align*} -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*} -\Delta v - \nu v = a(x)…

We examine the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=(1+|x|^2)^{\frac{\alpha}{2}} v,\\ -\Delta v=(1+|x|^2)^{\frac{\alpha}{2}} u^p, \end{cases} \quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$,…

In this paper, we consider the following nonlinear elliptic equation with gradient term: \[ \left\{ \begin{gathered} - \Delta u - \frac{1}{2}(x \cdot \nabla u) + (\lambda a(x)+b(x))u = \beta u^q +u^{2^*-1}, \hfill 0<u \in…

In this paper, we are mainly concerned with the physically interesting static Schr\"{o}dinger-Hartree-Maxwell type equations \begin{equation*} (-\Delta)^{s}u(x)=\left(\frac{1}{|x|^{\sigma}}\ast |u|^{p}\right)u^{q}(x)…

We consider the standing-wave problem for a nonlinear Schr\"{o}dinger equation, corresponding to the semilinear elliptic problem \begin{equation*} -\Delta u+V(x)u=|u|^{p-1}u,\ u\in H^1(\mathbb{R}^2), \end{equation*} where $V(x)$ is a…

In this paper we shall classify all positive solutions of $ \Delta u =a u^p$ on the upper half space $ H =\Bbb{R}_+^n$ with nonlinear boundary condition $ {\partial u}/{\partial t}= - b u^q $ on $\partial H$ for both positive parameters $a,…

In this paper, we consider the existence of solutions for the following fractional coupled Hartree-Fock type system \begin{align*} \left\{\begin{aligned} &(-\Delta)^s u+V_1(x)u+\lambda_1u=\mu_1(I_{\alpha}\star…

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation $$ -\Delta u-k^2 u =Q(x)|u|^{p-2}u,\qquad u \in W^{2,p}(\mathbb{R}^N) $$ with $N\geq 3$, $\frac{2(N+1)}{(N-1)}<…

We study the coupled Hartree system $$ \left\{\begin{array}{ll} -\Delta u+ V_1(x)u =\alpha_1\big(|x|^{-4}\ast u^{2}\big)u+\beta \big(|x|^{-4}\ast v^{2}\big)u &\mbox{in}\ \mathbb{R}^N,\\[1mm] -\Delta v+ V_2(x)v =\alpha_2\big(|x|^{-4}\ast…

This paper deals with a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta u+(V_1(x)+\lambda_1)u=\mu_1(|x|^{-4}*u^{2})u+\beta (|x|^{-4}*v^{2})u, \ \ &x\in R^N, -\Delta…

We consider positive solutions of the following elliptic Hamiltonian systems \begin{equation} \left\{ \begin{aligned} -\Delta u+u&=a(x)v^{p-1}~~~\text{in}~~A_R\\ -\Delta v+v&=b(x)u^{q-1}~~~\text{in}~~A_R~~~~~~~~~~~~~~~~~(0.1)\\ u,…

In this paper, we investigate the qualitative properties of positive solutions for the following two-coupled elliptic system in the punctured space: $$ \begin{cases} -\Delta u =\mu_1 u^{2q+1} + \beta u^q v^{q+1} \\ -\Delta v =\mu_2 v^{2q+1}…

In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: \begin{equation*}\begin{cases} -\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\dx\right)\Delta u+\lambda…

In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…