相关论文: Existence of a solution to a vector-valued Allen-C…

We obtain a sequence of solutions converging to zero for the Kirchhoff equation $$-\left( 1+\int_{\Omega}\left\vert \nabla u\right\vert^2\right) \Delta u+V(x)u=f(u)\text{,\qquad}u\in H_{0}^{1}(\Omega)$$ via truncating technique and a…

We construct weak solutions to the Navier-Stokes inequality, $$ u\cdot \left(\partial_t u -\nu \Delta u + (u\cdot \nabla) u +\nabla p \right) \leq 0 $$ in $\mathbb{R}^3$, which blow up at a single point $(x_0,T_0)$ or on a set $S \times…

The vector Riemann-Hilbert problem is analyzed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros…

In this paper, we consider the following linearly coupled Kirchhoff--Choquard system in $\mathbb{R}^3$: \begin{align*} \begin{cases} -\left(a_1 + b_1\int_{\mathbb{R}^3} |\nabla u|^2\,dx\right)\Delta u + V_1 u = \mu (I_{\alpha} * |u|^p)…

We consider the spatially $3$-D version of the following Keller-Segel-Navier-Stokes system with rotational flux $$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla…

We revisit the following fractional Schr\"{o}dinger equation \begin{align}\label{1a} \varepsilon^{2s}(-\Delta)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N, \end{align} where $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ denotes…

We study entire minimizers of the Allen-Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal…

In this paper we prove that Schr\"{o}dinger's equation with a Hamiltonian of the form $H=-\Delta+i(A \nabla + \nabla A) + V$, which includes a magnetic potential $A$, has the same dispersive and solution decay properties as the free…

We study the one-dimensional symmetry of solutions to the nonlinear Stokes equation $$ \begin{cases} -\Delta u+\nabla W(u)=\nabla p&\text{in }\mathbb{R}^d,\\ \nabla\cdot u=0&\text{in }\mathbb{R}^d, \end{cases} $$ which are periodic in the…

We study weak solutions to mean curvature flow satisfying Young's angle condition for general contact angles $\alpha \in (0,\pi)$. First, we construct BV solutions using the Allen-Cahn approximation with boundary contact energy as proposed…

This paper considers the fractional Schr\"{o}dinger equation \begin{equation}\label{abstract} (-\Delta)^s u + V(|x|)u-u^p=0, \quad u>0, \quad u\in H^{2s}(\R^N) \end{equation} where $0<s<1$, $1<p<\frac{N+2s}{N-2s}$, $V(|x|)$ is a positive…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

An exact solution of the three-dimensional spherical symmetry inverse-square-root potential $V\left( r\right) =-\frac{\alpha}{\sqrt{r}}$, including scattering and bound-state solutions, is presented.

We consider quasilinear elliptic problems of the form \[ -\operatorname{div}\big(\phi(|\nabla u|)\nabla u\big)+V(x)\phi (|u|)u=f(u)\qquad u\in W^{1,\Phi}(\mathbb{R}^{N}), \] where $\phi$ and $f$ satisfy suitable conditions. The positive…

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main…

We establish the existence of non-constant periodic solutions to the Lorentz force equation, where no scalar potential is needed to induce the electromagnetic field. Our results extend to cases where a possibly singular scalar potential is…

We consider the stationary nonlinear magnetic Choquard equation [(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\ast |u|^{p}) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where $A\ $is a real valued vector potential, $V$ is a real…

We are interested in the study of local and global minimizers for an energy functional of the type $$ \frac{1}{4} \iint_{\mathbb{R}^{2 N} \setminus \left( \mathbb{R}^N \setminus \Omega \right)^2} |u(x) - u(y)|^2 K(x - y) \, dx dy +…

In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with H\'enon-type weights \[ -\Delta u = |x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad { in } \Omega, \qquad u=v=0 {…

We discuss special k=sqrt{2m(E-V(x))/\hbar^2}=0 (i. e. zero-curvature) solutions of the one-dimensional Schrodinger equation in several model systems which have been used as idealized versions of various quantum well structures. We consider…