Singularly perturbed elliptic problems with nonautonomous asymptotically linear nonlinearities

Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

We consider the Laplace equation in a domain of $\mathbb{R}^n$, $n\ge 3$, with a small inclusion of size $ε$. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For $ε$ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions.

Pullback attractors for a singularly nonautonomous plate equation

We consider the family of singularly nonautonomous plate equation with structural damping \[ u_{tt} + a(t,x)u_{t} + (- Δ) u_{t} + (-Δ)^{2} u + λu = f(u), \] in a bounded domain $Ω\subset \R^n$, with Navier boundary conditions. When the nonlinearity $f$ is dissipative we show that this problem is globally well posed in $H^2_0(Ω) \times L^2(Ω)$ and has a family of pullback attractors which is upper-semicontinuous under small perturbations of the damping $a$.

Singularly perturbed biharmonic problems with superlinear nonlinearities

We are interested in finding a family of solutions to a singularly perturbed biharmonic equation which has a concentration behavior. The proof is based on variational methods and it is used a weak version of the Ambrosetti-Rabinowitz condition.

Note on semiclassical states for the Schrödinger equation with nonautonomous nonlinearities

We consider the following Schrödinger equation $$ - \hslash ^2 Δu + V(x)u = Γ(x) f(u) \quad \mathrm{in} \ \mathbb{R}^N, $$ where $u \in H^1 (\mathbb{R}^N)$, $u > 0$, $\hslash > 0$ and $f$ is superlinear and subcritical nonlinear term. We show that if $V$ attains local minimum and $Γ$ attains global maximum at the same point or $V$ attains global minimum and $Γ$ attains local maximum at the same point, then there exists a positive solution for sufficiently small $\hslash>0$.

非自治双曲问题的时间周期解的更高正则性：远离共振

Higher Regularity of Time-Periodic Solutions to Nonautonomous Hyperbolic Problems: Away from Resonances

该研究针对一维非自治双曲系统边值问题的时间周期解，探讨其更高正则性与非共振条件的关系，揭示非自治双曲PDE的共振特性，确定无需非共振条件的问题类并给出抽象正则性原理作为证明工具。

Multiple semiclassical states for fractional Schrodinger equations with asymptotically linear nonlinearities

In this paper, we consider the singularly perturbed fractional Schrödinger equation \begin{equation*} ε^{2α}(-Δ)^αu+V(x)u=f(u),\quad x\in \mathbb{R}^N, \end{equation*} where $ε>0$ is a small parameter, $α\in(0,1)$, $(-Δ)^α$ is the fractional Laplacian operator of order $α$, $V$ possesses global minimum points, and $f$ is asymptotically linear at infinity. We investigate the relationship between the number of positive solutions and the topology of the set where the potential $V$ attains its global minimum. We also construct multiple concentrating solutions if $V$ has several strict global minim…