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We investigate the following inhomogeneous nonlinear Schr\"odinger equation with focusing energy critical nonlinearity and a defocusing perturbation in the radial regime, $$ i \partial_t u +\Delta u =|x|^{-a} |u|^{p-2} u - |x|^{-b} |u|^{4-2b}u \quad \mbox{in} \quad \mathbb{R}_t \times \mathbb{R}_x^3, $$ where $0

We investigate the blow-up for fourth-order Schr\"odinger equation with a mas-critical focusing inhomogeneous nonliniearity. We prove the finite/infinite time blow-up of non-radial solutions with negative energy. Our result serves as a valuable complement to the existing literature, offering an improvement in our understanding of the subject matter.

We investigate a class of nonlinear Schr\"odinger equations with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, $$ i \partial_t u +\Delta u =|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2} |u|^{p_2-2}u \quad \mbox{in} \,\, \mathbb{R} \times \mathbb{R}^N, $$ where $N \geq 1$, $b_1, b_2>0$ and $p_1,p_2>2$. First, we establish the existence/nonexistence, symmetry, decay, uniqueness, non-degeneracy and instability of ground states. Then, we prove the scattering versus blowup below the ground state energy threshold. Our approach relies on Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. We also obtain an upper bound of the blow-up rate. The novelty here is that the equation does not enjoy any scaling invariance due to the presence of competing nonlinearties and the singular weights prevent the invariance by translation in the space variable. To the best of authors knowledge, this is the first time when inhomegenous NLS equation with a focusing leading order nonlinearity and a defocusing perturbation is investigated., Comment: 48 pages

This work studies the inhomogeneous Schr\"odinger equation $$ i\partial_t u-\mathcal{K}_{s,\lambda}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, $s\in\{1,2\}$, $N>2s$ and $\lambda>-\frac{(N-2)^2}{4}$. The linear Schr\"odinger operator reads $\mathcal{K}_{s,\lambda}:= (-\Delta)^s +(2-s)\frac{\lambda}{|x|^2}$ and the focusing source term is local or non-local $$F(x,u)\in\{|x|^{-2\tau}|u|^{2(q-1)}u,|x|^{-\tau}|u|^{p-2}(J_\alpha *|\cdot|^{-\tau}|u|^p)u\}.$$ The Riesz potential is $J_\alpha(x)=C_{N,\alpha}|x|^{-(N-\alpha)}$, for certain $0<\alpha0$ gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely $1+\frac{2(1-\tau)}N

This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad \mathcal K_\lambda=-\Delta+\frac\lambda{|x|^2}$$ in the energy space $H_\lambda^1:=\{f\in L^2,\quad\sqrt{\mathcal{K}_\lambda}f\in L^2\}$. In this paper, we develop a well-posedness theory and investigate the blow-up of solutions in $H_\lambda^1$. Furthermore we present a dichotomy between energy bounded and non-global existence of solutions under the ground state threshold. To this end, we use Caffarelli-Kohn-Nirenberg weighted interpolation inequalities and some equivalent norms considering $\mathcal K_\lambda$, which make it possible to control the non-linearity involving the singularity $|x|^{-\tau}$ as well as the inverse square potential. The novelty here is the investigation of the energy critical regime which remains still open and the challenge is to deal with three technical problems: a non-local source term, an inhomogeneous singular term $|\cdot|^{-\tau}$, and the presence of an inverse square potential., Comment: 24 pages

We consider the magnetic nonlinear inhomogeneous Schr\"odinger equation $$i\partial_t u -\left(-i\nabla+\frac{\alpha}{|x|^2}(-x_2,x_1)\right)^2 u =\pm|x|^{-\varrho}|u|^{p-1}u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^2,$$ where $\alpha\in\mathbb{R}\setminus\mathbb{Z},\,\varrho>0,\,p>1$. We prove a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground state threshold in the inter-critical regime. The scattering is obtained by using the new approach of Dodson-Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, {Proc. Am. Math. Soc.} (2017)). This method is based on Tao's scattering criteria and Morawetz estimates. The novelty here is twice: we investigate the case $\varrho\alpha\neq0$ and we consider general energy initial data (not necessarily radially symmetric)., Comment: We handle general initial data (not necessarily radially symmetric)

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-\Delta)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2

This manuscript proves the energy scattering of global solutions to a repulsive fourth-order generalized Hartree equation with non-radial data in the inter-critical regime. This work uses a new approach due to Dodson-Murphy [4] and extends the previous work [14] by removing the spherically symmetric assumption on the data., Comment: arXiv admin note: substantial text overlap with arXiv:2010.13315

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schr\"odinger equation $i \partial_t u + \Delta u = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. Our proofs are based on improved Gagliardo-Nirenberg inequalities, the Morawetz-Sobolev approach of Dodson and Murphy, radial Sobolev embeddings, and localized virial estimates., Comment: 41 pages, an error in the proof of the energy scattering has been fixed

This note studies the asymptotic behavior of global solutions to the fourth-order generalized Hartree equation $$i\dot u+\Delta^2 u\pm(I_\alpha*|u|^p)|u|^{p-2}u=0.$$ Indeed, for both attractive and repulsive sign, the scattering is obtained in the mass super-critical and energy sub-critical regimes, with radial setting.

This paper studies the inhomogeneous fractional Sch\"odinger equation $$i\dot u-(-\Delta)^s u=\pm(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u.$$ In the mass super-critical and energy sub-critical regimes, using a Gagliardo-Nirenberg adapted to the above problem, the standing waves give a sharp threshold of global existence versus finite time blow-up of solutions.

This paper studies the asymptotic behavior of global solutions to the generalized Hartree equation $$i\dot u+\Delta u+(I_\alpha *|\cdot|^b|u|^p)|x|^b|u|^{p-2}u=0 .$$ Indeed, using a new approach due to \cite{dm}, one proves the scattering of the above inhomogeneous Choquard equation in the mass-super-critical and energy sub-critical regimes with radial setting.

This note studies the asymptotic behavior of global solutions to the fourth-order Schr\"odinger equation $$i\dot u+\Delta^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson-Murphy \cite{dm}., Comment: arXiv admin note: substantial text overlap with arXiv:2010.07760

The initial value problem for some coupled nonlinear Schrodinger system with unbounded potential is investigated. In the defocusing case, global well-posedness is obtained. For the focusing sign, existence of global and non global solutions is discussed via potential well method. Moreover, existence of ground state and instability of standing waves are proved., Comment: arXiv admin note: text overlap with arXiv:1505.07506, arXiv:1505.07059, arXiv:1505.08101, arXiv:1505.08096

We investigate some focusing fourth-order coupled Schrodinger equations. Existence of ground state and global well-posedness are obtained. Moreover, the best constant of some Gagliardo-Nirenberg inequality is studied., Comment: arXiv admin note: substantial text overlap with arXiv:1505.07506

We investigate the initial value problem for some defocusing coupled nonlinear fourth-order Schrodinger equations. Global well-posedness and scattering in the energy space are obtained., Comment: arXiv admin note: substantial text overlap with arXiv:1505.07059

Some focusing coupled Schrodinger equations are investigated. First, existence of ground state is obtained. Second, global and non global existence of solutions are discussed via potential-well method. Finally, strong instability of standing waves is established.

The initial value problem for some defocusing coupled nonlinear Schrodinger equations is investigated. Global well-posedness and scattering are established., Comment: 19 pages