The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation $$ \Delta u+h_1e^u-h_2e^{-u}=0 \qquad \mathrm{in}~B_1\subset\mathbb R^2. $$ In particular, we get that the blow-up values are multiple of $8\pi.$ It generalizes the result of Jost, Wang, Ye and Zhou [20] where the extra assumption $h_1 = h_2$ is crucially used.

Let $T_{\varepsilon}$ be the lifespan for the solution to the Schrödinger equation on $\mathbb R^d$ with a power nonlinearity $\lambda |u|^{2\theta/d}u$ ($\lambda \in \mathbb C$, $0< \theta < 1$) and the initial data in the form $\varepsilon \varphi(x)$. We provide a sharp lower bound estimate for $T_{\varepsilon}$ as $\varepsilon \to +0$ which can be written explicitly by $\lambda$, $d$, $\theta$, $\varphi$ and $\varepsilon$. This is an improvement of the previous result by H. Sasaki [Adv. Diff. Eq., 14 (2009), 1021-1039].

Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the discontinuity of the flow map in the super-critical range.

In this paper, we consider a system of semilinear equations involving the fractional Laplacian in the Euclidean space $\mathbb{R}^n$: \begin{equation*} \begin{cases} (-\Delta)^{\alpha/2}u(x)=f(x_n)v^p(x)\\ (-\Delta)^{\alpha/2}v(x)=g(x_n)u^q(x) \end{cases} \end{equation*} in the subcritical case $1 < p,q\le \frac{n+\alpha}{n-\alpha}$ where $\alpha \in (0,\,2)$. Instead of investigating the above system directly, we discuss its equivalent integral system: \begin{equation*} \begin{cases} u(x)=\int_{\mathbb{R}^n} G_{\infty}(x,y)f(y_n)v^p(y)dy\\ v(y)=\int_{\mathbb{R}^n} G_{\infty}(x,y)g(x_n)u^q(x)dx , \end{cases} \end{equation*} where $G_{\infty}(x, y)$ is the Green's function associated with the fractional Laplacian in $\mathbb{R}^n$. Under natural structure condition on $f$ and $g$, we indicate the nonexistence of the positive solutions to the above integral system according to the method of moving spheres in integral form and the classic Hardy-Littlewood-Sobolev inequality.

In this paper, we study an eigenvalue problem with piecewise constant coefficients on thin domains with Neumann boundary condition, and we analyze the asymptotic behavior of each eigenvalue as the domain degenerates into a certain hypersurface being the set of discontinuities of the coefficients. We show how the discontinuity of the coefficients and the geometric shape of the interface affect the asymptotic behavior of the eigenvalues by using a variational approach.

The classical eigenvalue theory for second-order ordinary differential equations (ODE) describes the spatial oscillation of strings whose distributions of masses are absolutely continuous. For general distributions of masses, including completely singular ones, the spatial oscillation can be explained using measure differential equations (MDE). In this paper we will study weighted eigenvalue problems for second-order MDE with general distributions or measures. It will be shown that the numbers of weighted eigenvalues depend on measures and may be finite. Furthermore, it will be proved that weighted eigenvalues and eigenfunctions are completely continuous in measures, i.e., when measures are convergent in the weak$^*$ topology, these eigen-pairs are strongly convergent. The present paper and the work of Meng and Zhang (Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Equations, 254 (2013), 2196-2232, have given an extension of the classical Sturm-Liouville theory to the measure version of ODE.