Related papers: Qualitative analysis for Moser-Trudinger nonlinear…

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality \[ \sup \left\{\int_\Omega \exp{\left(\alpha\,|u|^{\frac{N}{N-s}}\right)}\,\bigg|\,u \in…

For the Schr\"odinger equation, $ (i \partial_t + \Delta) u = 0 $ on a torus, an arbitrary non-empty open set $ \Omega $ provides control and observability of the solution: $ \| u |_{t = 0} \|_{L^2 (\T^2)} \leq K_T \| u \|_{L^2 ([0,T]…

We consider the $L^2$-critical nonlinear Schr\"odinger equation (NLS) with the delta potential $$i\partial_tu +\partial^2_x u + \mu \delta u +|u|^{4}u=0, \, \, t\in \R, \, x\in \R , $$ where $ \mu \in \R$, and $\delta$ is the Dirac delta…

We investigate the focusing inhomogeneous nonlinear biharmonic Schr\"odinger equation \[ i\partial_t u + \Delta^2 u - |x|^{-b}|u|^p u = 0 \quad \text{on } \mathbb{R} \times \mathbb{R}^N, \] in the energy-critical regime, $p = \frac{8 -…

We consider the following nonlinear Schr\"{o}dinger equation with an inverse potential: \[ i\frac{\partial u}{\partial t}+\Delta u+|u|^{\frac{4}{N}}u\pm\frac{1}{|x|^{2\sigma}}\log|x|u=0 \] in $\mathbb{R}^N$. From the classical argument, the…

Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation $$(-\Delta)^\frac n2 u_k=\lambda_k…

We consider the problem of Ambrosetti-Prodi type \begin{equation}\label{0}\quad\begin{cases} \Delta u + e^u = s\phi_1 + h(x) &\hbox{in} \Omega, u=0 & \hbox{on} \partial \Omega, \end{cases} \nonumber \end{equation} where $\Omega$ is a…

We study existence, regularity, and qualitative properties of solutions to the system \[ -\Delta u = |v|^{q-1} v\quad \text{ in }\Omega,\qquad -\Delta v = |u|^{p-1} u\quad \text{ in }\Omega,\qquad \partial_\nu u=\partial_\nu v=0\quad \text{…

In this paper, we study the following fractional nonlinear Schr\"odinger system $$ \left\{% \begin{array}{ll} (-\Delta)^s u +u=\mu_1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u,~~x\in \R^N,\vspace{2mm}\\ (-\Delta)^s v +v=\mu_2 |v|^{2p-2}v+\beta…

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$…

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon…

Let $\Omega$ be a bounded domain in $\mathbb R^{N}$, $N\geq3$ with smooth boundary, $a>0, \lambda>0$ and $0<\delta<3$ be real numbers. Define $2^*:=\displaystyle\frac{2N}{N-2}$ and the characteristic function of a set $A$ by $\chi_A$. We…

We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equations possessing weakly dissipative structure in one space dimension. We show that the small data solution decays like $O((\log t)^{-1/4})$ in $L^2$ as…

In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in $H^1$. The well-posedness theory in $H^1$ has been intensively…

We consider the equation $- \Delta u+V(x)u- k(\Del(|u|^{2}))u=g(x,u), u>0, x \in {\BR}^2,$ where $V:{\BR}^2\to {\BR}$ and $g:{\BR}^2 \times {\BR}\to {\BR}$ are two continuous $1-$periodic functions. Also, we assume $g$ behaves like $\exp…

We consider the following nonlinear Schr\"odinger equations with critical growth: \begin{equation} - \Delta u + V(|y|)u=u^{\frac{N+2}{N-2}},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \end{equation} where $V(|y|)$ is a bounded positive radial…

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and $\delta(x)$ be the distance of a point $x\in \Omega$ to the boundary. We study the positive solutions of the problem $\Delta u +\frac{\mu}{\delta(x)^2}u=u^p$ in $\Omega$, where $p>0,…

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $\Delta u+g(u)=0$. Our result applies in particular to the double power non-linearity where…

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -\Delta u=(|x|^{-(n-2)}\ast u^{p-\epsilon})u^{p-1-\epsilon}\quad…