Related papers: Nondegeneracy of positive solutions to nonlinear H…

We consider the singular elliptic problem of the form \[ -\Delta u + V(x)u = \mathcal{B}(x)|u|^{2^*-2}u + \frac{\mathcal{A}(x)}{|u|^{2^*}u}, \qquad u\in H^1(M), \] where the coefficients are allowed to have low regularity. Under natural…

In this paper we consider nonlinear elliptic PDEs of the type $$-\Delta_p u+a(x)|u|^{p-2}u=|u|^{p^*-2}u \qquad \mbox{ in }\Omega,$$ where $1<p<N$ and $p^*=Np/(N-p)$ is the critical Sobolev exponent, and allowing the asymptotic behavior of…

We study the critical Neumann problem \begin{equation*} \begin{cases} -\Delta u = |u|^{2^*-2}u &\text{in }\Sigma_\omega,\\ \quad\frac{\partial u}{\partial\nu}=0 &\text{on }\partial\Sigma_\omega, \end{cases} \end{equation*} in the unbounded…

In this paper, we are interested in some problems related to the following biharmonic hartree equation \begin{equation*} \Delta^{2} u=(|x|^{-\alpha}\ast |u|^{p})u^{p-1},\sp \text{in}\quad\R^N. \end{equation*} where…

Derivation of an exact, general solution to Newell-Whitehead-Segel transient, nonlinear partial differential equation is provided for one to three dimensional cases, also, arbitrary power of nonlinearity.

It was recently shown, that in a class of tensor-multi-scalar theories of gravity with a nontrivial target space metric, there exist scalarized neutron star solutions. An important property of these compact objects is that the scalar charge…

We consider the defocusing, energy subcritical wave equation $\partial_t^2 u - \Delta u = -|u|^{p-1} u$ in 4 to 6 dimensional spaces with radial initial data. We define $w=r^{(d-1)/2} u$, reduce the equation above to one-dimensional…

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $$ -\nabla \cdot \left(|x|^{2a} \nabla u\right) + \omega u=|u|^{p-2}u \quad \mbox{in} \,\, \R^d, $$…

In this article, we study the following nonlinear Choquard equation with singular nonlinearity \begin{equation*} \quad -\De u = \la u^{-q} + \left( \int_{\Om}\frac{|u|^{2^*_{\mu}}}{|x-y|^{\mu}}\mathrm{d}y \right)|u|^{2^*_{\mu}-2}u, \quad…

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with…

We prove new results concerning the nonlinear scalar field equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = g(u)&\quad \hbox{in }\mathbb{R}^N,\; N\geq 3, u\in H^1(\mathbb{R}^N)& \end{array} \right. \end{equation*} with a…

The aim of this paper is to prove radial symmetry results for positive weak solutions with finite energy to the following quasilinear doubly critical system \begin{equation} \begin{cases} -\Delta_p u\,=\gamma \frac{u^{p-1}}{|x|^p} +…

We investigate the existence and multiplicity of positive solutions to the problem \begin{equation} \begin{cases} \begin{aligned} - \Delta_{\gamma} u &= \lambda u^{p} + u^{-\delta} &\quad \text{in } \Omega, \quad u &= 0 &\quad \text{on }…

A nonlinear extension of Schr\"odinger's wave equation is proposed that ensures non-signaling by keeping linear the evolution of \textit{coordinate-diagonal} elements of the density matrix. The equation contains a negative kinetic energy…

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 prove uniqueness and nondegeneracy of positive solutions to the following Kirchhoff equations with critical growth \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u=u^{5}, & u>0 & \text{in…

We consider a time-dependent one-dimensional nonlinear Schroedinger equation with a symmetric potential double well represented by two delta interactions. Among our results we give an explicit formula for the integral kernel of the unitary…

We consider the following prescribed scalar curvature equations in ${\mathbb{R}}^N$ $$ - \Delta u =K(|y|)u^{2^*-1},\quad u>0 \quad \mbox{in} \quad {\mathbb{R}}^N, \quad u \in D^{1, 2}({\mathbb{R}}^N), $$ where $K(r)$ is a positive function,…

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may…

We consider the problem of describing all non-negative integer solutions to a linear congruence in many variables. This question may be reduced to solving the congruence $x_1 + 2x_2 + 3x_3 + ... + (n-1)x_{n-1} \equiv 0 \pmod n$ where values…