Related papers: Normalized solutions for a Kirchhoff type equation…

In this paper, we are concerned with normalized solutions in $H_{r}^{1}(\mathbb{R}^{3}) \times H_{r}^{1}(\mathbb{R}^{3})$ for Hartree-Fock type systems with the form \be\lab{ Hartree-Fock} \left\{ \begin{array}{ll} -\Delta u +\alpha \phi…

In this paper, we consider the existence of normalized solutions for the following $p$-Laplacian equation \begin{equation*} \left\{\begin{array}{ll} -\Delta_{p}u-V(x)\lvert u\rvert^{p-2}u+\lambda\lvert u\rvert^{p-2}u=\lvert…

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 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 this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)\Delta…

In this paper, we consider the following nonlinear Kirchhoff type problem: \[ \left\{\begin{array}{lcl}-\left(a+b\displaystyle\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\ u\in…

By introducing some new tricks, we prove that the nonlinear problem of Kirchhoff-type \begin{equation*} \left\{ \begin{array}{ll} -\left(a+b\int_{\R^3}|\nabla u|^2\mathrm{d}x\right)\triangle u+V(x)u=f(u), & x\in \R^3; u\in H^1(\R^3),…

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…

We are concerned with solutions of the following quasilinear Schr\"odinger equations \begin{eqnarray*} -{\mathrm{div}}\left(\varphi^{2}(u) \nabla u\right)+\varphi(u) \varphi^{\prime}(u)|\nabla u|^{2}+\lambda u=f(u), \quad x \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 this article, we study the existence of normalized ground state solutions for the following biharmonic nonlinear Schr\"{o}dinger equation with combined nonlinearities \begin{equation*} \Delta^2u=\lambda u+\mu|u|^{q-2}u+|u|^{p-2}u,\quad…

We study normalised solutions of the stationary Gross-Pitaevskii-Poisson (GPP) equation with a defocusing local nonlinear term, $$-\Delta u+\lambda u+|u|^2u =(I_\alpha*|u|^2)u\quad\text{in $\mathbb R^3$},\qquad\int_{\mathbb…

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation $$ \left\{\begin{array}{ll} -(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu…

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$…

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 following logarithmic Schr\"{o}dinger system: $$\left\{\begin{align} \ &\ -\Delta u_1+\omega_1u_1=\mu_1 u_1\log u_1^2+\frac{2p}{p+q}|u_2|^{q}|u_1|^{p-2}u_1,\\ \ &\ -\Delta u_2+\omega_2u_2=\mu_2 u_2\log…

In this paper, we prove the existence of normalized solutions for the following Schr\"odinger equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u-\lambda u=f(u), & x\in \R^N, \int_{\R^N}u^2\mathrm{d}x=c \end{array} \right.…

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 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 }…

For $N\ge 3$ and $2<p<N$, we find normalised solutions to the equation \begin{align*} -\Delta_p u+(1+V(x))|u|^{p-2}u+\lambda u&=|u|^{q-2}u\qquad\text{in $\mathbb{R}^N$}\\ \|u\|_2&=\rho \end{align*} in the mass supercritical and Sobolev…