Vol 87, No 1 (2023)

In this paper, we are concerned with the following Schrödinger–Poisson system$$\begin{cases}-\Delta u+u+\lambda\phi u= Q(x)|u|^{4}u+\mu\dfrac{|x|^\beta}{1+|x|^\beta}|u|^{q-2}u&in \mathbb{R}^3, -\Delta \phi=u^{2} &in \mathbb{R}^3, \end{cases}$$where $0< \beta<3$,  $60$ are real parameters. By the variationalmethod and the Nehari method, we obtain that the system has $k$ positivesolutions.

We study the eigenvalues of the Neumann and Dirichlet boundary-value problems in a two-dimensionaldomain containing several small, of diameter $O(\varepsilon)$, inclusions of large “density”$O(\varepsilon^{-\gamma})$, $\gamma\geq2$, that is, the “mass” $O(\varepsilon^{2-\gamma})$ of each of them is comparable ($\gamma=2$) or much bigger ($\gamma>2$) than that of the embracingmaterial. We construct a model of such spectral problems on concentrated masses which (the model)provides an asymptotic expansions of the eigenvalues with remainders of power-law smallnessorder $O(\varepsilon^{\vartheta})$ as $\varepsilon\to+0$ and $\vartheta\in(0,1)$. Besides,the correction terms are real analytic functions of the parameter $|{\ln \varepsilon}|^{-1}$. A “far-field interaction”of the inclusions is observed at the levels $|{\ln \varepsilon}|^{-1}$or $|{\ln \varepsilon}|^{-2}$. The results are obtained with the help of the machineryof weighted spaces with detached asymptotics and also by using weighted estimates of solutions to limit problemsin a bounded punctured domain and in the intact plane.

The paper gives a detailed study of long-time dynamics generated byweakly damped wave equations in bounded 3D domains where the dampingcoefficient depends explicitly on time and may change sign. It is shown thatin the case, where the non-linearity is superlinear, the considered equationremains dissipative if the weighted mean value of the dissipation rateremains positive and that the conditions of this type are not sufficient inthe linear case. Two principally different cases are considered. In thecase when this mean is uniform (which corresponds to deterministicdissipation rate), it is shown that the considered system possesses smoothuniform attractors as well as non-autonomous exponential attractors. In thecase where the mean is not uniform (which corresponds to the randomdissipation rate, for instance, when this dissipation rate is generated bythe Bernoulli process), the tempered random attractor is constructed. Incontrast to the usual situation, this random attractor is expected to haveinfinite Hausdorff and fractal dimensions. The simplified model exampledemonstrating infinite-dimensionality of the random attractor is alsopresented.