Related papers: Ground state solution of a Kirchhoff type equation…

We consider the $L^2$-critical nonlinear Schr\"odinger equation (NLS) with the delta potential $$i\partial_tu +\partial^2_x u + \mu \delta u +|u|^{4}u=0, \, \, t\in \R, \, x\in \R , $$ where $ \mu \in \R$, and $\delta$ is the Dirac delta…

We establish a new estimate for the Ginzburg-Landau energies $E_{\epsilon}(u)=\int_M\frac{1}{2}|du|^2+\frac{1}{4\epsilon^2}(1-|u|^2)^2$ of complex-valued maps $u$ on a compact, oriented manifold $M$ with $b_1(M)\neq 0$, obtained by…

This paper is dedicated to the blow-up solution for the divergence Schr\"{o}dinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where…

We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality $$\iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{|u(x)|^p\,|u(y)|^p}{|x - y|^{N-\alpha}} dx\, dy\le C\Big(\int_{{\mathbb R}^N}|u|^2 dx\Big)^{p\theta}…

We prove a theorem for the growth of the energy of bounded, globally minimizing solutions to a class of semilinear elliptic systems of the form $\Delta u=\nabla W(u)$, $x\in \mathbb{R}^n$, $n\geq 2$, with $W:\mathbb{R}^m\to \mathbb{R}$,…

We show that the finite time type II blow up solutions for the energy critical nonlinear wave equation \[ \Box u = -u^5 \] on $\mathbb R^{3+1}$ constructed by Krieger-Schlag-Tataru are stable along a co-dimension one Lipschitz manifold of…

In this paper, we are interested in the following planar Choquard equation \begin{equation*} \begin{cases} -\Delta u=\displaystyle\left(\int\limits_{\Omega}\frac{u^{p+1}(y)}{|x-y|^\alpha}dy\right)u^{p},\quad u>0,\ \ &\mbox{in}\ \Omega,…

We investigate the following Kirchhoff-type biharmonic equation \begin{equation}\label{pr} \left\{ \begin{array}{ll} \Delta^2 u+ \left(a+b\int_{\mathbb{R}^N}|\nabla u|^2d x\right)(-\Delta u+V(x)u)=f(x,u),\quad x\in \mathbb{R}^N,\\ u\in…

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…

By means of exact diagonalization we study the low-energy states of seven electrons in the lowest Landau level which are confined by a cylindric external potential modelling the rest of a macroscopic system and thus controlling the filling…

We investigate the point singularity of very weak solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations. More precisely, assume that the solution $(\mathbf{u},\mathbf{B})$ in the punctured ball $B_2\setminus \{0\}$ satisfies…

In this paper, we study the discrete fractional logarithmic Kirchhoff equation $$ \left(a+b \int_{\mathbb{Z}^d}|\nabla^s u|^{2} d \mu\right) (-\Delta)^s u+h(x) u=|u|^{p-2}u \log u^{2}, \quad x\in \mathbb{Z}^d, $$ where $a,\,b>0$ and…

In this paper, we study the static Born-Infeld equation $$ -\mathrm{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)=\sum_{k=1}^n a_k\delta_{x_k}\quad\mbox{in }\mathbb R^N,\qquad \lim_{|x|\to\infty}u(x)=0, $$ where $N\ge3$,…

We consider the generalized Choquard equation describing trapped electron gas in 3 dimensional case. The study of orbital stability of the energy minimizers (known as ground states) depends essentially in the local uniqueness of these…

We investigate minimizers defined on a bounded domain in $\mathbb{R}^2$ for the Maier--Saupe Q--tensor energy used to characterize nematic liquid crystal configurations. The energy density is singular, as in Ball and Mujamdar's modification…

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling type equation $\triangle^2u+\lambda\triangle u-k^2u=0$ in a bounded analytic domain…

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_\Omega |\nabla u|^2 \dif x +b\right)\Delta u = \lambda u+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $\lambda \in…

In this paper, we study the multiplicity and concentration of the positive solutions to the following critical Kirchhoff type problem: \begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d} x\right)\Delta u…

We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schr\"odinger equations of the…

We provide a sufficient condition for the existence of a positive solution to $-\Delta u+V(|x|) u=u^p$ in $B_1$, when p is large enough. Here $B_1$ is the unit ball of $R^n$, n greater or equal to 2, and we deal both with Neumann and…