American Institute of Mathematical Sciences

This note studies the fourth-order Choquard equation

\begin{document}$ i\dot u+\Delta^2 u\pm (I_\alpha *|u|^p)|u|^{p-2}u = 0 . $\end{document}

In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.

In this article, we provide existence results to the following nonlocal equation

\begin{document}$ \begin{align*} \begin{cases} (-\Delta)_p^{s} u = g(x,u),\;u>0\; \text{in}\; \Omega,\\ u = 0 \; \text{in}\; \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*}\quad\quad(P_ \lambda)$ \end{document}

where \begin{document}$ (-\Delta)_{p}^{s} $\end{document} is the fractional \begin{document}$ p $\end{document}-Laplacian operator. Here \begin{document}$ \Omega \subset \mathbb R^N $\end{document} is a smooth bounded domain, \begin{document}$ s\in(0,1) $\end{document}, \begin{document}$ p>1 $\end{document} and \begin{document}$ N>sp $\end{document}. We establish existence of at least one weak solution for \begin{document}$ (P_ \lambda) $\end{document} when \begin{document}$ g(x,u) = f(x)u^{-q(x)} $\end{document} and existence of at least two weak solutions when \begin{document}$ g(x,u) = \lambda u^{-q(x)}+ u^{r} $\end{document} for a suitable range of \begin{document}$ \lambda>0 $\end{document}. Here \begin{document}$ r\in(p-1,p_{s}^{*}-1) $\end{document} where \begin{document}$ p_s^{*} $\end{document} is the critical Sobolev exponent and \begin{document}$ 0<q \in C^1(\bar{ \Omega}) $\end{document}.

In this paper we prove existence and concentration results for a family of nodal solutions for a some quasilinear equation with subcritical growth, whose prototype is

\begin{document}$ -\Delta_{p}u- \Delta_{q}u+V( x)(|u|^{p-2}u+|u|^{q-2}u) = f(u) \quad \mbox{in} \ \mathbb{R}^{N}. $\end{document}

Each nodal solution changes sign exactly once in \begin{document}$ \mathbb{R}^{N} $\end{document} and has an exponential decay at infinity. Here we use variational methods and Del Pino and Felmer's technique [10] in order to overcome the lack of compactness.

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [25] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

In this paper, we study the existence of \begin{document}$ L^2 $\end{document}-normalized solutions for the following 3-coupled nonlinear Schrödinger equations in \begin{document}$ [H_r^1( \mathbb{R}^N)]^3 $\end{document},

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u_i = \lambda_i u_i+\mu_i|u_i|^{p_i-2}u_i+\beta r_i|u_i|^{r_i-2}\big(\sum\limits_{j\neq i}|u_j|^{r_j}\big)u_i,\\ |u_i|_2^2 = a_i, \quad i, j = 1,2,3, \end{cases} \end{equation*} $\end{document}

where \begin{document}$ \mu_i, \beta $\end{document} and \begin{document}$ a_i $\end{document} are given positive constants, \begin{document}$ \lambda_i $\end{document} appear as unknown parameters, and \begin{document}$ H_r^1( \mathbb{R}^N) $\end{document} denotes the radial subspace of Hilbert space \begin{document}$ H^1( \mathbb{R}^N) $\end{document}. For \begin{document}$ p_i, r_i $\end{document} satisfying \begin{document}$ L^2 $\end{document}-subcritical or \begin{document}$ L^2 $\end{document}-supercritical conditions, we obtain positive solutions of this system using variational methods and perturbation methods.

We consider the Klein-Gordon system posed in an inhomogeneous medium \begin{document}$ \Omega $\end{document} with smooth boundary \begin{document}$ \partial \Omega $\end{document} subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood \begin{document}$ \omega $\end{document} of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [5]. Although the present problem has some similarity to the reference [6] it is important to mention that due to the Kelvin-Voigt dissipation character associated with the nonlinearity of the problem the approach used is completely new, which is the main purpose of this paper.

In this paper, we consider the existence of quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs near the equilibrium point for most parameter values.

In this paper we study the asymptotic behaviour of solutions for a non-local non-autonomous scalar quasilinear parabolic problem in one space dimension. Our aim is to give a fairly complete description of the forward asymptotic behaviour of solutions for models with Kirchhoff type diffusion. In the autonomous case we use the gradient structure, symmetry properties and comparison results to obtain a sequence of bifurcations of equilibria, analogous to what is seen in the local diffusivity case. We provide conditions so that the autonomous problem admits at most one positive equilibrium and analyse the existence of sign changing equilibria. Also using symmetry and the comparison results (developed here) we construct what is called non-autonomous equilibria to describe part of the asymptotics of the associated non-autonomous non-local parabolic problem.

We are concerned with the Cauchy problem of the inelastic Boltzmann equation for soft potentials, with a Laplace term representing the random background forcing. The inelastic interaction here is characterized by the non-constant restitution coefficient. We prove that under the assumption that the initial datum has bounded mass, energy and entropy, there exists a weak solution to this equation. The smoothing effect of weak solutions is also studied. In addition, it is shown the solution is unique and stable with respect to the initial datum provided that the initial datum belongs to \begin{document}$ L^{2}(R^{3}) $\end{document}.

This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that \begin{document}$ \|\rho_0\|_{L^\infty}+\|\nabla d_0\|_{L^3} $\end{document} is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.

We investigate the following gauged nonlinear Schrödinger equation

\begin{document}$ \begin{equation*} \begin{cases} -\Delta u+\omega u+\lambda\bigg(\dfrac{h_{u}^{2}(|x|)}{|x|^{2}}+ \int_{|x|}^{+\infty}\dfrac{h_{u}(s)}{s}u^{2}(s)ds\bigg)u = f(u) \ \ \ \ \ \mbox{in}\ \mathbb{R}^{2},\\ u\in H_r^1(\mathbb{R}^{2}), \end{cases} \end{equation*} $\end{document}

where \begin{document}$ \omega,\lambda>0 $\end{document} and \begin{document}$ h_{u}(s) = \frac{1}{2}\int_{0}^{s}ru^{2}(r)dr $\end{document}. When \begin{document}$ f $\end{document} has exponential critical growth, by using the constrained minimization method and Trudinger-Moser inequality, it is proved that the equation has a ground state radial sign-changing solution \begin{document}$ u_{\lambda} $\end{document} which changes sign exactly once. Moreover, the asymptotic behavior of \begin{document}$ u_{\lambda} $\end{document} as \begin{document}$ \lambda\rightarrow0 $\end{document} is analyzed.

In this paper, using the method of moving planes combined with integral inequality to handle the fractional Laplacian system, we prove Liouville type theorems of nonnegative solution for the nonlinear system.

In this paper, we consider nonlinear problems involving nonlocal Monge-Ampère operators. By using a sliding method, we establish monotonicity of positive solutions for nonlocal Monge-Ampère problems both in an infinite slab and in an upper half space. During this process, an important idea we applied is to estimate the singular integrals defining the nonlocal Monge-Ampère operator along a sequence of approximate maximum points. It allows us to assume weaker conditions on nonlinear terms. Another idea is to employ a generalized average inequality which plays an important role and greatly simplify the process of the sliding.

We employ Clarkson's inequality to deduce that each extremal of Morrey's inequality is axially symmetric and is antisymmetric with respect to reflection about a plane orthogonal to its axis of symmetry. We also use symmetrization methods to show that each extremal is monotone in the distance from its axis and in the direction of its axis when restricted to spheres centered at the intersection of its axis and its antisymmetry plane.