We study the large time behavior of solutions for the nonlinear Schrödinger equation with subcritical nonlinearities $$ \sum _{n=0,1}|u|^{1+p-n}u^n $$ with $0 < p < \frac{2}{d}$. We prove that there is no scattering nor modified scattering of solutions for the nonlinear Schrödinger equation.

We consider a nonlinear Neumann boundary value problem which is derived for the Antarctic Circumpolar Current. By the theory of topological degree, we prove the existence results for the problem with semilinear oceanic vorticity term. We also construct the approximate solutions for such a nonlinear model.

The paper establishes well-posedness for the Cauchy problem associated with the Zakharov-Kutnesov equation in the cylinder for certain generalized dispersion. The main ingredient in this work is a localized Strichartz estimate and a compactness argument.

In this article, we study some optimal design problems related to nonstandard growth eigenvalues ruled by the $g-$Laplacian operator. More precisely, given $\Omega\subset \mathbb{R}^n$ and $\alpha,c>0$, we consider the optimization problem $\inf \{ \lambda_\Omega(\alpha,E)\colon E\subset \Omega, |E|=c \}$, where $\lambda_\Omega(\alpha,E)$ is related to the first eigenvalue to $$ -{\rm div\,}(g( |\nabla u |)\tfrac{\nabla u}{|\nabla u|}) + g(u)\tfrac{u}{|u|}+ \alpha \chi_E g(u)\tfrac{u}{|u|} \quad \text{ in }\Omega $$ subject to Dirichlet, Neumann or Steklov boundary conditions. We analyze existence of an optimal configuration, symmetry properties of them, and the asymptotic behavior as $\alpha$ approaches $+\infty$.

In this paper, we are concerned with normalized solutions of the fractional critical Choquard equation with a local perturbation and prescribed mass. For the $L^2$-subcritical case, we study the multiplicity of normalized solutions by applying the truncation technique, the concentration-compactness principle and the genus theory. For the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map and using the concentration-compactness principle.

We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level, we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.

This paper is concerned with the Sobolev norm growth for the solution to the one-dimensional cubic fourth-order Schrödinger equation. Applying Tao's $[k;Z]$-multiplier method, we gain some bilinear estimates. Then we show the solution satisfies $$ \|u(t)\|_{H^s}\leq \|u(\tau)\|_{H^s}+ C\|u(\tau)\|_{H^s}^{1-\delta}, \qquad\delta^{-1}=(s-2)^+ $$ and then derive a polynomial upper bound of time $t$.

We consider the existence and stability of traveling wave solutions for Nicholson's blowflies equation with degenerate $p$-Laplacian diffusion and time delay. Existence results are proved by the monotone iteration method. Furthermore, we prove the existence and regularity of original solution for the time-delayed degenerate Nicholson's blowflies equation via compactness analysis. Finally, we prove the stability and exponential convergence rate of traveling waves by an approximated weighted energy method.

We introduce and study a Kelvin-type transform in the anisotropic setting. In particular, we deal with semilinear and quasilinear anisotropic elliptic problems in the entire space.