Related papers: Sign-changing bubble tower solutions for sinh-Pois…

We construct infinite time bubble tower solutions to the critical wave maps equation taking values in the two-sphere. More precisely, for any integers $k\geq3$ and $J\geq1$, we construct a solution that is global in one time direction, has…

Topological transitions in bubbling half-BPS Type IIB geometries with SO(4) x SO(4) symmetry can be decomposed into a sequence of n elementary transitions. The half-BPS solution that describes the elementary transition is seeded by a phase…

We study the boundary regularity of solutions to the porous medium equation $u_t = \Delta u^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the…

In this paper, we consider the fractional heat equation with critical exponent in $\mathbb{R}^n$ for $n>6s,s\in(0,1),$ \begin{equation*} u_t=-(-\Delta)^su+|u|^{\frac{4s}{n-2s}}u,\quad (x,t)\in \mathbb{R}^n\times\mathbb{R}. \end{equation*}…

We consider the sinh-Poisson equation $$(P)_\lambda\quad -\Delta u=\la\sinh u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega,$$ where $\Omega$ is a smooth bounded domain in $\rr^2$ and $\lambda$ is a small positive parameter. If…

We prove the existence of two smooth families of unbounded domains in $\mathbb{R}^{N+1}$ with $N\geq1$ such that \begin{equation} -\Delta u=\lambda u\,\, \text{in}\,\,\Omega, \,\, u=0,\,\,\partial_\nu…

The present paper is concerned with the Cauchy-Dirichlet problem for fractional (and non-fractional) nonlinear diffusion equations posed in bounded domains. Main results consist of well-posedness in an energy class with no sign restriction…

We consider the supercritical Lane-Emden problem $$(P_\eps)\qquad -\Delta v= |v|^{p_\eps-1} v \ \hbox{in}\ \mathcal{A} ,\quad u=0\ \hbox{on}\ \partial\mathcal{A} $$ where $\mathcal A$ is an annulus in $\rr^{2m},$ $m\ge2$ and…

In this paper, we study the existence of localized sign-changing (or nodal) solutions for the following nonlinear Schr\"odinger-Poisson system \begin{equation*} \begin{cases} -\varepsilon^2 \Delta u+V(x)u+\phi…

In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial…

We study the spherical mean transform on $\rN^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of 1+1-dimension hyperbolic equations, which provide…

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Delta u_2=|u_1|^{q_\epsilon-1}u_1, \…

We study the critical problem {equation} {{array}{ll} \Delta ^{2}u=u^{\frac{N+4}{N-4}} & {in}\Omega\setminus \bar{B(\xi_0,\varepsilon)},\medskip u>0&{in}\Omega\setminus \bar{B(\xi_0,\varepsilon)},\medskip u=\Delta u=0 & {on}\partial (\Omega…

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -\Delta u = \lambda u + |u|^{2^* -2}u & \hbox{in}\ \Omega\\ u=0 & \hbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in…

A new protocol for linearization of the Poisson-Boltzmann equation is proposed and the resultant electrostatic equation coincides formally with the Debye-Huckel equation, the solution of which is well known for many electrostatic problems.…

We study the global and local existence and uniqueness of solutions to the Navier-Stokes equations with anisotropic viscosity in a bounded cylindrical domain $Q=\Omega\times (0,1)$, where $\Omega$ is a star-shaped domain in $R^2$. In this…

This paper is concerned with the existence of sign-changing solutions to non local Kirchhoff type problems of the form \begin{equation}\label{s}\tag{S} -\Big(a+b\int_\Omega|\nabla u|^2dx\Big)\Delta u=f(x,u)\, \text{ in }\Omega,\quad\quad…

Let $(M, g)$ be a compact Riemannian manifold with boundary. The Yamabe problem concerning the existence of a metric conformally equivalent to $g$ having constant scalar curvature on $M$ and constant mean curvature on its boundary is…

In this article we study the existence of sign changing solution of the following p-fractional problem with concave-critical nonlinearities: \begin{eqnarray*} (-\Delta)^s_pu &=& \mu |u|^{q-1}u + |u|^{p^*_s-2}u \quad\mbox{in}\quad \Omega,…

We show that signed weak solutions to obstacle problems for porous medium type equations with Cauchy-Dirichlet boundary data are continuous up to the parabolic boundary, provided that the obstacle and boundary data are continuous. This…