Related papers: Removable singularity of positive solutions for a …
We consider the uniqueness of the following positive solutions of anisotropic elliptic equation: \begin{equation} \nonumber \left\{ \begin{aligned} -\Delta^F_p u&=u^q \quad \text{in} \quad \Omega, u&=0 \quad \text{on} \quad \partial \Omega,…
We study solutions to conformally invariant equations with isolated singularties.
In this paper we give a new, less restrictive condition for removability of singular sets, $E$, of smooth solutions to the m-Hessian equation (and also for more general fully nonlinear elliptic equations) in $\Omega \setminus E$, $\Omega…
We study the existence of positive bound states for the nonlinear elliptic system \[ \begin{cases} - \Delta u_i + \lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i & \text{in $\Omega$} \\ u_1 =\cdots = u_d=0 & \text{on $\partial \Omega$},…
Coupled nonlinear Schrodinger equations (CNLS) with an external elliptic function potential model a quasi one--dimensional interacting two-component Bose-Einstein condensate trapped in a standing light wave. New families of stationary…
We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…
In this paper, we prove a sharp uniqueness result for the singular Schr\"odinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent…
The one-dimensional Schr\"{o}dinger equation with the singular harmonic oscillator is investigated. The Hermiticity of the operators related to observable physical quantities is used as a criterion to show that the attractive or repulsive…
In this paper, we consider the existence and asymptotic behavior on mass of the positive solutions to the following system: \begin{equation}\label{eqA0.1}\nonumber \begin{cases} -\Delta u+\lambda_1u=\mu_1u^3+\alpha_1|u|^{p-2}u+\beta…
We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and…
We show removability of half-line singularities for viscosity solutions of fully nonlinear elliptic PDEs which have classical density and a Jacobi inequality. An example of such a PDE is the Monge-Amp\`ere equation, and the original proof…
We consider a Hamiltonian system of free boundary type, showing first uniform bounds and existence of solutions and of the free boundary. Then, for any smooth and bounded domain, we prove uniqueness of positive solutions in a suitable…
We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain…
The homogeneous Lippmann-Schwinger integral equation is solved in momentum space by using confining potentials. Since the confining potentials are unbounded at large distances, they lead to a singularity at small momentum. In order to…
We consider positive solutions of the following elliptic Hamiltonian systems \begin{equation} \left\{ \begin{aligned} -\Delta u+u&=a(x)v^{p-1}~~~\text{in}~~A_R\\ -\Delta v+v&=b(x)u^{q-1}~~~\text{in}~~A_R~~~~~~~~~~~~~~~~~(0.1)\\ u,…
We study properties of positive functions satisfying (E) --$\Delta$u + u p -- M |$\nabla$u| q = 0 is a domain $\Omega$ or in R N + when p > 1 and 1 < q < min{p, 2}. We concentrate our research on the solutions of (E) vanishing on the…
We build infinitely-many non-radial positive solutions to the Schr\"odinger system \begin{equation*} \left\{\begin{aligned} &-\Delta u_1+u_1=u_1^{{\mathfrak p} }-\Lambda u_1^{a_1} u_2^{a_2}\ \hbox{in}\ \mathbb R^N\\ &-\Delta…
We consider the existence and multiplicity of positive solutions for the following critical problem with logarithmic term: \begin{equation*}\label{eq11}\left\{ \begin{array}{ll} -\Delta u={\mu\left|u\right|}^{{2}^{\ast }-2}u+\nu…
We investigate the positive solutions of the semilinear elliptic equation \begin{align*} \sum^{N}_{i=1}\left(-\partial_{ii}\right)^{s}u=u^{p} \end{align*} with one-dimensional symmetric $2s$-stable operators. Firstly, in the whole space…
Coupled nonlinear Schrodinger systems describe some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive media in optics and Bose-Einstein condensates. In this paper, we study the existence of…