相关论文: Normalized solutions for the NLS equation with pot…

This paper is concerned with the existence of solutions to the problem $$-\left(a+ b\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx \right)\Delta u +V(x)u+\lambda u = |u|^{p-2}u,\ \ x \in \mathbb{R}^{N},\ \ \lambda \in \mathbb{R}^{+} $$ where $a,…

In this paper, we consider the existence of normalized solution to the following Kirchhoff equation with mixed Choquard type nonlinearities: \begin{equation*} \begin{cases} -\left(a + b \int_{\mathbb{R}^3} |\nabla u|^2 \, dx\right) \Delta u…

We study the stationary nonlinear Schr\"odinger equation \begin{equation}-\Delta u+V(x)u+\lambda u=|u|^{q-2}u,\quad u \in H^1(\mathbb{R}^N), \quad N \geq 2\end{equation} where $V \in L^{\infty}(\mathbb{R}^N)$ is a radial potential. In the…

We consider the existence of solutions $(\lambda_1,\lambda_2, u, v)\in \mathbb{R}^2\times (H^1(\mathbb{R}^N))^2$ to systems of coupled Schr\"odinger equations $$ \begin{cases} -\Delta u+\lambda_1 u=\mu_1 u^{p-1}+\beta r_1…

In this paper, we study normalized solutions for the following critical Schr\"odinger-Bopp-Podolsky system: $$-\Delta u + q(x)\phi u = \lambda u + |u|^{p-2}u + \bigl(I_\alpha * |u|^{3+\alpha}\bigr)|u|^{1+\alpha}u,\quad \text{in }…

\noindent We are concerned with positive normalized solutions $(u,\lambda)\in H^1(\mathbb{R}^2)\times\mathbb{R}$ to the following semi-linear Schr\"{o}dinger equations $$ -\Delta u+\lambda u=f(u), \quad\text{in}~\mathbb{R}^2, $$ satisfying…

In this paper we study the normalized solutions of the following critical growth Choquard equation with mixed local and non-local operators: \begin{equation*} \begin{array}{rcl} -\Delta u +(-\Delta)^s u & = & \lambda u +\mu |u|^{p-2}u…

This work examines a quasilinear Schr\"odinger-Poisson system involving a critical nonlinearity, expressed as \[ -\Delta u + \phi u + \lambda u = |u|^{q-2} u + |u|^4 u, \quad x \in \Omega_r, \] \[ -\Delta \phi - \varepsilon^4 \Delta_4 \phi…

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…

The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated…

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…

This paper is concerned with the following focusing biharmonic HLS upper critical Hartree equation with a local perturbation $$ \begin{cases} {\Delta}^2u-\lambda u-\mu|u|^{p-2}u-(I_\alpha*|u|^{4^*_\alpha})|u|^{4^*_\alpha-2}u=0\ \ \mbox{in}\…

In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schr\"odinger equation $$(-\Delta)^s u + \mu u = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ under the mass constraint…

In this paper we investigate the existence of the positive solutions for the following nonlinear Schr\"odinger equation $$ -\triangle u+V(x)u=K(x)|u|^{p-2}u\ {in}\ \mathbb{R}^N $$ where $V(x)\sim a|x|^{-b}$ and $K(x)\sim \mu|x|^{-s}$ as…

We investigate the existence of normalized solutions for the following nonlinear fractional Choquard equation: $$ (-\Delta)^s u+V(\epsilon x)u=\lambda u+\left(I_\alpha *|u|^q\right)|u|^{q-2} u+\left(I_\alpha *|u|^p\right)|u|^{p-2} u, \quad…

In the present work we are concerned with the existence of normalized solutions to the following Schr\"odinger-Poisson System $$ \left\{ \begin{array}{ll} -\Delta u + \lambda u + \mu (\ln|\cdot|\ast |u|^{2})u = f(u) \textrm{ \ in \ }…

In this paper, we study the existence and multiplicity of the normalized solutions to the following quasi-linear problem \begin{equation*} -\Delta u-\Delta(|u|^2)u+\lambda u=|u|^{p-2}u+\tau|u|^{q-2}u, \text{ in }\mathbb{R}^N,~ 1\leq N\leq4,…

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…

A regularized $\alpha-$system of the Nonlinear Schr\"{o}dinger Equation (NLS) with $2\sigma$ nonlinear power in dimension $N$ is studied. We prove existence and uniqueness of local solution in the case $1 \le \sigma <\frac{4}{N-2}$ and…

In this paper, we study the existence and non-existence of normalized solutions to the lower critical Choquard equation with a local perturbation \begin{equation*} \begin{cases} -\Delta u+\lambda u=\gamma…