We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.

In this paper, we deal with the following critical fractional Schrödinger-Poisson system without subcritical perturbation $$ \begin{cases} (-\Delta)^{s} u + V(x)u + K(x)\phi u=|u|^{2_{s}^{*}-2}u, & x\in\mathbb{R}^{3},\\ (-\Delta)^{t}\phi=K(x)u^{2}, & x\in\mathbb{R}^{3}, \end{cases} $$ where $s\in(\frac{3}{4},1),$ $t\in(0,1)$, $2^{\ast}_{s}=\frac{6}{3-2s}$ is the critical Sobolev exponent. When $V(x)$ is positive bounded from below, combining with variation methods and Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this system. The results obtained in this paper extend and improve some recent works in which potential function $V(x)$ may vanish at the infinity.

In this paper, we provide an existence result for a scalar differential equation ruled by a nonlinearity near resonance presenting a one-sided superlinear condition. The result is achieved by the introduction of a suitable Landesman-Lazer type condition. The main theorem can be extended also to the case of equations with a singularity. The proof is given by using a shooting method technique.

We discover a new minimality property of the absolute minimizers of supremal functionals, whose variational problems are also known as $L^\infty$ variational problems. In particular for every minimizer $v$ of the quasi-convex functional $$ {\rm ess.sup}\,{\Omega}H(x,Dv(x)), $$ we consider the set $$ \mathcal{A}(v)=\{x: H (x,Dv(x))={\rm ess.sup}\,{\Omega}H(x,Dv(x))\}, $$ suitably defined. If $u$ is an absolute minimizer, we give a structure result for $\mathcal{A}(u)$ and we show that $\mathcal{A}(u)\subset\mathcal{A}(v)$ for every minimizer $v$.

This paper completes some previous studies by several authors on the finite time extinction for nonlinear Schrödinger equation when the nonlinear damping term corresponds to the limit cases of some ``saturating non-Kerr law'' $F(|u|^2)u=\frac{a}{{\varepsilon}+(|u|^2)^\alpha}u,$ with $a\in{\mathbb{C}},$ ${\varepsilon}\ge0,$ $2\alpha=(1-m)$ and $m\in[0,1).$ Here, we consider the sublinear case $0 < m < 1$ with a critical damped coefficient: $a\in{\mathbb{C}}$ is assumed to be in the set $ D(m)=\big\{z\in{\mathbb{C}}; \; {\mathrm{Im}}(z) > 0 \text{ and } 2\sqrt m{\mathrm{Im}}(z)=(1-m){\mathrm{Re}}(z)\big\} . $ Among other things, we know that this damping coefficient is critical, for instance, in order to obtain the monotonicity of the associated operator (see the paper by Liskevich and Perel'muter [16] and the more recent study by Cialdea and Maz'ya [14]). The finite time extinction of solutions is proved by a suitable energy method after obtaining appropriate a priori estimates. Most of the results apply to non-necessarily bounded spatial domains.

There are two flaws in the third part [3] of our study on the Navier--Stokes equations in a curved thin domain. The first one is that the form of Ladyzhenskaya's inequality on a closed surface taken from the arXiv version [2] of the second part of our study is incorrect. The second one is that a wrong claim is used in the proofs of lemmas in [3, Section 6.3] which give estimates for the weighed potential part of a tangential surface vector field. This erratum corrects these flaws.