Woo-Cheol Choi

In this paper, we study standing waves for pseudo-relativistic nonlinear Schrödinger equations. In the first part, we find ground state solutions. We also prove that they have one sign and are radially symmetric. The second part is devoted to take nonrelativistic limit of the ground state solutions in H1(ℝn) space.

Abstract We fill a gap in the proofs in the previous works (Wu X, Mu, X. Int J. Robust Nonlin Control. 2020;30:3753Ű3772; Zhang Z, Lunze J, Wang L. Int J Control. 2020;93:1005‐1014; Zhang Z, Wang L. J Robust Nonlin Control. 2018;28:4175Ű4187; Dai, M‐Z, Zhang C, Leung H, Dong P, Li B. IEEE Trans Syst, Man, Cybern: Syst. doi:10.1109/TSMC.2021.3119670) for the consensus using the integral‐based event‐triggered controls. More precisely, it was inferred for a Lyapunov function that is uniformly bound

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators $L_K+V$ with nonnegative potentials $V∈ L^q_{\rm{loc}}(\mathbb{R}^n)$ for $q>\frac{n}{2s}$ with $0 < s < 1$ and $n>2s$; that is to say, we obtain the existence of a fundamental solution $\mathfrak{e}_V$ for $L_K+V$ satisfying $\begin{equation*}\bigl(L_K+V\bigr)\mathfrak{e}_V = \delta _0\,\,\text{ in $\mathbb{R}^n$ }\end{equation*}$ in the distribution sense, where $\delta _0$ denotes the Dirac

In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity \[ \left(\sqrt{-\hbar^2c^2 Δ+m^2c^4} - mc^2 \right) u + μu = \mathcal{N}(u), \] where $c$ denotes the speed of light. We prove that the ground states of this equation converges to the ground state of its nonrelativistic counterpart \[ -\frac{\hbar^2}{2m}Δu + μu = \mathcal{N}(u) \] with an explicit convergence rate $1/c^2$ in arbitrary o

In this paper we study standing waves for pseudo-relativistic nonlinear Schrödinger equations. In the first part we find ground state solutions. We also prove that they have one sign and are radially symmetric. The second part is devoted to take nonrelativistic limit of the ground state solutions in $H^1 (\mathbb{R}^n)$ space.

In this paper, we consider nonlocal Schrödinger equations with certain potentials $V∈{\rm{RH}}^q$($q>\frac{n}{2s}>1$ and $0<s <1$) of the form \begin{document}$\begin{equation*}L_K u+V u = f\,\,\text{ in }\; \mathbb{R}^n \end{equation*}$ \end{document} where $L_K$ is an integro-differential operator. We denote the solution of the above equation by $\mathcal{S}_V f: = u$, which is called the inverse of the nonlocal Schrödinger operator $L_K+V$ with potential $V$; that is, $\mathcal{S}

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left( \sqrt{-c^2Δ+ m^2 c^4}-mc^2\right) u + μu = |u|^{p-1}u\quad \textrm{in}~\mathbb{R}^n~(n \geq 2)$$ involving an $H^{1/2}$-critical/supercritical power-type nonlinearity, i.e., $p \geq \frac{n+1}{n-1}$. We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is $H^{1/2}$-critical/supercritical bu

Abstract In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left( \sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = \vert u \vert^{p-1}u\quad {\rm in}~{\open R}^n~(n \ges 2) $$ involving an H 1/2 -critical/supercritical power-type nonlinearity, that is, p ⩾ (( n + 1)/( n − 1)). We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is H 1/2 -crit