相关论文: Construction of solutions to a nonlinear critical …

In this paper, we consider the following PDE involving two Sobolev-Hardy critical exponents, \label{0.1} {& \Delta u + \lambda\frac{u^{2^*(s_1)-1}}{|x|^{s_1}} + \frac{u^{2^*(s_2)-1}}{|x|^{s_2}} =0 \text{in} \Omega, & u=0 \qquad \text{on}…

We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v,…

In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -\Delta_{p_1}u_1 & =-|u_1|^{p_1-2}u_1 \quad && \text{in } \Omega,\newline -\Delta_{p_2}u_2 & =-|u_2|^{p_2-2}u_2 \quad && \text{in }…

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - \Delta u &= f(|x|,u,v)\quad\text{in}\R^n, - \Delta v &=…

We study asymptotic behaviour of positive ground state solutions of the nonlinear Schr\"odinger equation $$ -\Delta u+ u=u^{2^*-1}+\lambda u^{q-1} \quad {\rm in} \ \ \mathbb{R}^N, $$ where $N\ge 3$ is an integer, $2^*=\frac{2N}{N-2}$ is the…

We prove the uniqueness for the positive solutions of the following elliptic systems: \begin{eqnarray*} \left\{\begin{array}{ll} - \lap (u(x)) = u(x)^{\alpha}v(x)^{\beta} - \lap (v(x)) = u(x)^{\beta} v(x)^{\alpha} \end{array} \right.…

In this paper, we consider the nonlocal elliptic problems in $\mathbb{R}^{N}$, which involve finite many critical exponents. By using endpoint refined Hardy--Sobolev inequality, fractional Coulomb--Sobolev space and variational method, we…

This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where…

In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Delta v=|x|^{\alpha} |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ \Omega, \end{equation*}where $\Omega$…

We consider the semilinear elliptic boundary value problem \[ -\Delta u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on }\partial\Omega, \] in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for supercritical…

We study the Sobolev critical Schr\"odinger-Bopp-Podolsky system \begin{gather*} -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3, -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, \end{gather*}…

We revisit the following nonlinear critical elliptic equation \begin{equation*} -\Delta u+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about the…

Consider the equation \begin{equation*} -\Delta_p u =\lambda |u|^{p-2}u+\mu|u|^{q-2}u+|u|^{p^\ast-2}u\ \ {\rm in}\ \R^N \end{equation*} under the normalized constraint $$\int_{ \R^N}|u|^p=c^p,$$ where $-\Delta_pu={\rm div} (|\nabla…

We consider the following mixed local and non-local critical elliptic equation: \begin{equation*}\label{0.1} \left\{ \begin{array}{lll} -\Delta u+(-\Delta)^su=\lambda h u^{p}+u^{2^*-1}, &\text{in}\,\, \mathbb{R}^n, u>0, &\text {in} \,\,…

In this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem \begin{equation}\label{1.1ab} \left\{ \begin{aligned} -\Delta {u}-\frac{1}{2}(x \cdot{\nabla u})&= \lambda{|u|^{{2}^{*}-2}u}+{\mu…

We are concerned with a semi-linear elliptic equation on a smooth bounded domain $\Omega$ of $\mathbb{R}^n,\,n\geq 5,$ which involves a critical nonlinearity and a linear term of the form $K(x)u^{(n+2)/(n-2)}$ and $\mu u,$ respectively. By…

In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -\Delta u= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~\Omega},\\[1mm] u>0,~ &{\text{in}~\Omega},\\[1mm] u=0, &{\text{on}~\partial…

We consider the critical Choquard system with both linear and nonlinear couplings $-\Delta v_1 + \mu_1 v_1 = ( I_\omega * |v_1|^{2_\omega^*} ) |v_1|^{2_\omega^* -2} v_1 + \theta p( I_\omega * |v_2|^q)|v_1|^{p-2} v_1 + \varepsilon v_2, \quad…

In this paper, we study the following Schr\"odinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -\Delta u+V(x)u+\lambda u=|u|^{q-2}u+\beta f(u), \\ & \int |u|^2dx=\Theta, \end{aligned}…

We consider the problem $(P)$, $$ -\Delta u =c(x)u+\mu|\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega),$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $\mu>0, \, c \in…