Related papers: Regularity of the Scattering Matrix for Nonlinear …

We analyze the shape of radial second Dirichlet eigenfunctions of fractional Schr\"odinger type operators of the form $(-\Delta)^s +V$ in the unit ball $B$ in $\mathbb{R}^N$ with a nondecreasing radial potential $V$. Specifically, we show…

We prove scattering for the radial nonlinear Klein-Gordon equation $ \partial_{tt} u - \Delta u + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) \in H^{s} \times H^{s-1} $, $ 1 > s > 1- \frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4…

We consider the inhomogeneous nonlinear Schr\"odinger equation $$ i u_t +\Delta u+|x|^{-b}|u|^\alpha u = 0, $$ where $\frac{4-2b}{N}<\alpha<\frac{4-2b}{N-2}$ (when $N=2$, $\frac{4-2b}{N}<\alpha<\infty$) and $0<b<\min\{N/3,1\}$. For a radial…

We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue…

In this paper, we study the scattering for the nonlinear beam equation $u_{tt}+\Delta^2u+mu+\mu |u|^{p-1}u=0$. Our results include two aspects. In the defocusing case we show that the scattering holds for $d=1$, which extends the result in…

Using the Fredholm theory of the linear time-dependent Schr\"odinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schr\"odinger problem $$ (D_t + \Delta + V) u = N[u], \quad…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schr\"odinger) equation $(\Delta + k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued…

We consider an elliptic Kolmogorov equation lambda u - Ku =f in a convex subset C of a separable Hilbert space X. We prove maximal Sobolev regularity of its weak solution, when lambda >0 and f is in L^2(C,nu), where nu is the log-concave…

We consider an eigenvalue problem for the generalized nonlinear Schr\"{o}dinger type operator with the Robin boundary condition as given below. \begin{equation*} \label{ab-Robin p-Laplace evp with potential term_intro} \left\{ \begin{split}…

In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $\Omega=\mathcal O\times\mathbb R$…

For $p>2$, we consider the quasilinear equation $-\Delta_p u+|u|^{p-2}u=g(u)$ in the unit ball $B$ of $\mathbb R^N$, with homogeneous Neumann boundary conditions. The assumptions on $g$ are very mild and allow the nonlinearity to be…

We consider the focusing inhomogeneous biharmonic nonlinear Schr\"odinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + \Delta^2 u - |x|^{-b}|u|^{\alpha}u=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data…

This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -\Delta_{p}u -\Delta_{q}u = f(x) u^{-\delta},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*}…

We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator…

We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness…

In this paper, we prove the decay and scattering in the energy space for nonlinear Schr\"odinger equations with regular potentials in $\Bbb R^d$ namely, $i{\partial _t}u + \Delta u - V(x)u + \lambda |u|^{p - 1}u = 0$. We will prove decay…

In this paper we study the semilinear elliptic problem $$ -\Delta u -k^2u=Q|u|^{p-2}u\quad\text{ in }\mathbb{R}^2, $$ where $k>0$, $p\geq 6$ and $Q$ is a bounded function. We prove the existence of real-valued $W^{2,p}$-solutions, both for…

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $\Delta^2 u -\beta \Delta u + \alpha u= \Gamma|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $\Gamma$. Using…

Let $\Delta_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:\Delta u + \lambda u =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate…