In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional Nonlinear Schrödinger Equation (NLS),

\begin{equation*}\mathrm{i} \, \partial_t u=(-\Delta)^s u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^N,\end{equation*}

where $N \geq 2$, $1/2 \lt s \lt 1$, and $0 \lt \sigma \lt 2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data.

It was proved in [11, J. Funct. Anal., 2020] that the Cauchy problem for some Oldroyd-B model is well-posed in $\dot{B}^{d/p-1}_{p,1}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,1}(\mathbb{R}^d)$ with $1\leq p \lt 2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\dot{B}^{d/p-1}_{p,r}(\mathbb{R}^d) \times \dot{B}^{d/p}_{p,r}(\mathbb{R}^d)$ with $1\leq p\leq \infty$ and $1 \lt r\leq\infty$ due to the lack of continuous dependence of the solution.

We are concerned with the following coupled Schrödinger system with a Hardy potential in the critical case

\begin{align*}\begin{cases}-\Delta u_{i}-\frac{\lambda_{i}}{|x|^2}u_{i}=|u_i|^{2^*-2}u_i+\sum_{j\neq i}^{3}\beta_{ij}|u_{j}|^{\frac{2^*}{2}}|u_i|^{\frac{2^*}{2}-2}u_i, ~x\in \mathbb{R}^N,\\u_i\in D^{1,2}(\mathbb{R}^N),\,\, N\geq 3,\,\, i=1,2,3,\end{cases}\end{align*}

$2^*=\frac{2N}{N-2}$

$\lambda_i\in (0,\Lambda_N), \Lambda_N:= \frac{(N-2)^2}{4}$

$\beta_{ij}=\beta_{ji}$

$\beta_{ij} \gt 0$

$\beta_{i_{1}j_{1}} \gt 0$

$\beta_{i_{2}j_{2}} \lt 0$

$(i_{1}, j_{1})$

$(i_{2}, j_{2})$

$\beta_{ij}\ge0$

$N\ge5$

$\beta_{ij} \gt 0$

$N=3,4$

$N=3, 4$

$N\geq 5$

$N\geq 5$

$N\ge5$

We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non-linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\mathcal{T})$, where ${\mathcal{T}}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.

Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular, we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing–Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterization of algebraically special Riemannian metrics of type D in dimensions higher than four.

We investigate a free energy functional that arises in aggregation-diffusion phenomena modelled by nonlocal interactions and local repulsion on the hyperbolic space ${\mathbb H}^n$. The free energy consists of two competing terms: an entropy, corresponding to slow nonlinear diffusion, that favours spreading, and an attractive interaction potential energy that favours aggregation. We establish necessary and sufficient conditions on the interaction potential for ground states to exist on the hyperbolic space ${\mathbb H}^n$. To prove our results, we derived several Hardy–Littlewood–Sobolev (HLS)-type inequalities on general Cartan–Hadamard manifolds of bounded curvature, which have an interest in their own.

We consider the drifting p-Laplace operator

\begin{equation*}\Delta_{p,v}u=e^{-v} \text{div}\,(e^v |\nabla u|^{p-2}\nabla u)\end{equation*}

and discuss generalized weighted Hardy-type inequalities associated with the measure $d\mu=e^{v(x)}dx$. As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term

\begin{equation*}-\Delta_{p,v} u\ge c(x) u^{p-1}+B \frac{|\nabla u|^p}{u} \quad \text{in}\ \Omega,\end{equation*}

where Ω is a bounded or an unbounded exterior domain in ${\mathbb{R}}^N$, $N \gt p \gt 1$, $B+p-1 \gt 0$, as well as of the non-autonomous quasilinear elliptic problem

\begin{equation*}-\Delta_{p,v} u+b(x)|\nabla u|^{p-1} \ge c(x) u^{p-1}\quad \text{in}\ \Omega,\end{equation*}

with general weights $b\ge0$ and c > 0. Liouville-type results are also discussed for a class of higher order differential equations.

We investigate the pullback measure attractors for non-autonomous stochastic p-Laplacian equations driven by nonlinear noise on thin domains. The concept of complete orbits for such systems is presented to establish the structures of pullback measure attractors. We first present some essential uniform estimates, as well as the existence and uniqueness of pullback measure attractors. A novel technical proof method is shown to overcome the difficulty of the estimates of the solutions in $W^{1,p}$ on thin domains. Then, we prove the upper semicontinuity of these measure attractors as the $(n + 1)$-dimensional thin domains collapse onto the lower n-dimensional space.

This paper develops methods for simplifying systems of partial differential equations (PDEs) that have families of conservation laws which depend on arbitrary functions of the independent or dependent variables. Cases are identified in which such methods can be combined with reduction using families of symmetries to give a multiple reduction; this is analogous to the double reduction of order for ordinary differential equations (ODE) with variational symmetries. Applications are given, including a widely used class of pseudoparabolic equations and several mean curvature equations.

This overview discusses the inverse scattering theory for the Kadomtsev–Petviashvili II equation, focusing on the inverse problem for perturbed multi-line solitons. Despite the introduction of new techniques to handle singularities, the theory remains consistent across various backgrounds, including the vacuum, 1-line and multi-line solitons.

In this paper, we study the Cauchy problem for pseudo-parabolic equations with a logarithmic nonlinearity. After establishing the existence and uniqueness of weak solutions within a suitable functional framework, we investigate several qualitative properties, including the asymptotic behaviour and blow-up of solutions as $t\to +\infty$. Moreover, when the initial data are close to a Gaussian function, we prove that these weak solutions exhibit either super-exponential growth or super-exponential decay.

Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we investigate the global (higher-order) Sobolev regularity of weak solutions to the fractional Dirichlet problem

\begin{equation*}\begin{cases}(-\Delta)^su=f \ \ & \text{in}\ \ \Omega,\\u=0 \ \ & \text{in}\ \ \mathbb{R}^n\setminus\Omega.\end{cases}\end{equation*}

Precisely, we prove that there exists a positive constant $\varepsilon\in(0,s]$ depending on n, s, and the Lipschitz constant of Ω such that, for any $t\in[\varepsilon,\min\{1+\varepsilon,2s\})$, when $f\in L^q(\Omega)$ with some $q\in(\frac{n}{2s-t},\infty]$, the weak solution u satisfies

\begin{equation*}\|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for all $p\in[1,\frac{1}{t-\varepsilon})$. In particular, when Ω is a bounded C1 domain or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the aforementioned global regularity estimates hold with $\varepsilon=s$ and they are sharp in this case. Moreover, if Ω is a bounded $C^{1,\kappa}$ domain with $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, we further show the global BMO-Sobolev regularity estimate

\begin{equation*}\left\|(-\Delta)^{\frac{s}{2}}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}+\left\|\nabla^{s}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for some $q\in(\frac{n}{s},\infty]$, which is sharp in the sense that the BMO norm can not be improved to the $L^\infty$ norm.

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analysing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker–Planck equation can be approximated using the Keller–Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker–Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller–Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker–Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.

Here we consider the following fractional Hamiltonian system

\begin{equation*}\begin{cases}\begin{aligned}(-\Delta)^{s} u&=H_v(u,v) \;\;&&\text{in}~\Omega,\\(-\Delta)^{s} v&=H_u(u,v) &&\text{in}~\Omega,\\u &= v = 0 &&\text{in} ~ \mathbb{R}^N\setminus\Omega,\end{aligned}\end{cases}\end{equation*}

where $s\in (0,1)$, $N \gt 2s$, $H \in C^1(\mathbb{R}^2, \mathbb{R})$, and $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain. To apply the variational method for this problem, the key question is to find a suitable functional setting. Instead of usual fractional Sobolev spaces, we use the solution space of $(-\Delta)^{s}u=f\in L^r(\Omega)$ for $r\ge 1$, for which we show the (compact) embedding properties. When H has subcritical and superlinear growth, we construct two frameworks, respectively with the interpolation space method and the dual method, to show the existence of nontrivial solution. As byproduct, we revisit the fractional Lane–Emden system, i.e. $H(u, v)=\frac{1}{p+1}|u|^{p+1}+\frac{1}{q+1}|v|^{q+1}$, and consider the existence, uniqueness of (radial) positive solutions under subcritical assumption.

This paper is the latter part of a series of our studies on the concentration and oscillation analysis of semilinear elliptic equations with exponential growth $e^{u^p}$. In the first one [17], we completed the concentration analysis of blow-up positive solutions in the supercritical case p > 2 via a scaling approach. As a result, we detected infinite sequences of concentrating parts with precise quantification. In the present paper, we proceed to our second aim, the oscillation analysis. Especially, we deduce an infinite oscillation estimate directly from the previous infinite concentration ones. This allows us to investigate intersection properties between blow-up solutions and singular functions. Consequently, we show that the intersection number between blow-up and singular solutions diverges to infinity. This leads to a proof of infinite oscillations of bifurcation diagrams, which ensures the existence of infinitely many solutions. Finally, we also remark on infinite concentration and oscillation phenomena in the limit cases $p\to2^+$ and $p\to \infty$.

We prove the convergence of a Wasserstein gradient flow of a free energy in inhomogeneous media. Both the energy and media can depend on the spatial variable in a fast oscillatory manner. In particular, we show that the gradient-flow structure is preserved in the limit, which is expressed in terms of an effective energy and Wasserstein metric. The gradient flow and its limiting behavior are analysed through an energy dissipation inequality. The result is consistent with asymptotic analysis in the realm of homogenisation. However, we note that the effective metric is in general different from that obtained from the Gromov–Hausdorff convergence of metric spaces. We apply our framework to a linear Fokker–Planck equation, but we believe the approach is robust enough to be applicable in a broader context.

We focus on the existence of ground state solutions to the following class of elliptic equations

\begin{equation*}-\Delta_{\mathbb{B}^N} u+V(x)u=f(u) \quad \hbox{in} \quad \mathbb{B}^N,\end{equation*}

where $\mathbb{B}^N$ is the disc model of the Hyperbolic space and $\Delta_{\mathbb{B}^N}$ denotes the Laplace–Beltrami operator with $N \geq 2$, $V:\mathbb{B}^N \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ are continuous functions that satisfy some technical conditions. With different types of the potential V, by introducing some new tricks handling the hurdle that the Hyperbolic space is not a compact manifold, we are able to obtain at least a positive ground state solution using variational methods.

As some applications for the methods adopted above, we derive the existence of normalized solutions to the elliptic problems

\begin{equation*}\left\{\begin{aligned}&-\Delta_{\mathbb{B}^N} u+V(x)u=\mu u+f(u) \quad \hbox{in} \quad \mathbb{B}^N,\\&\int_{\mathbb{B}^N}|u|^{2}\,dV_{{\mathbb{B}^N}}=a^{2},\end{aligned}\right.\end{equation*}

where a > 0, $\mu\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and f is a continuous function that fulfils the L2-subcritical or L2-supercritical growth. We do believe that it seems the first results to deal with normalized solutions for the Schrödinger equations in the Hyperbolic space.

We investigate radial and non-radial solutions to a class of (p, q)-Laplace equations involving weights. More precisely, we obtain existence and multiplicity results for nontrivial nonnegative radial and non-radial solutions, which extend results in the literature. Moreover, we study the non-radiality of minimizers in Hénon type (p, q)-Laplace problems and symmetry-breaking phenomena.

Dedicated to Professor Pavel Drábek on the occasion of his seventieth birthday

In this paper, we deal with the following nonlinear time-harmonic Maxwell’s equations

\begin{equation*}\nonumber\begin{cases} \nabla \times(\nabla \times u_1)=\mu_1|u_1|^4u_1+\beta|u_1||u_2|^3u_1 & \text {in } \mathbb{R}^3, \\ \nabla \times(\nabla \times u_2)=\mu_2|u_2|^4u_2+\beta|u_1|^3|u_2|u_2 & \text {in } \mathbb{R}^3, \end{cases}\end{equation*}

$\nabla\times$

$\mathbb{R}^3$

$\mu_1,\mu_2 \gt 0$

$\beta\in\mathbb{R}\backslash\{0\}$

$\beta\in\mathbb{R}\backslash\{0\}$

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide proof by employing Kato’s method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

In this paper, we study the following high-order elliptic equation involving the GJMS operator:

\begin{align*}\alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}.\end{align*}

We establish that if α > 1 and $n\geq3$ or if $\alpha\in (1-\epsilon_0, 1)$ with $n=2m\geq4$, then $v_{\alpha}\equiv0$. As an application, we present a new proof of the classical Beckner inequality.