Related papers: The nodal set of solutions to some elliptic proble…

In this paper we prove existence of least energy nodal solutions for the Hamiltonian elliptic system with H\'enon-type weights \[ -\Delta u = |x|^{\beta} |v|^{q-1}v, \quad -\Delta v =|x|^{\alpha}|u|^{p-1}u\quad { in } \Omega, \qquad u=v=0 {…

We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that…

In this paper, we establish optimal a priori $C^{1,\alpha}$ regularity estimates for the ratio $w = v/u$ of two solutions to the same elliptic equation $-\operatorname{div}(A \nabla u )=0$ with Lipschitz coefficients $A$, under the…

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…

We study some properties of positive solutions to the higher order conformally invariant equation with a singular set $$ (-\Delta)^m u = u^{\frac{n+2m}{n-2m}} ~~~~~~ \textmd{in} ~ \Omega \backslash \Lambda, $$ where $\Omega \subset…

We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-\Delta)^s u+\epsilon u &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \]…

We investigate the point singularity of very weak solutions $(\mathbf{u},\mathbf{B})$ to the stationary MHD equations. More precisely, assume that the solution $(\mathbf{u},\mathbf{B})$ in the punctured ball $B_2\setminus \{0\}$ satisfies…

This study examines nonnegative solutions to the problem \begin{equation*}\left\{\arraycolsep=1.5pt \begin {array}{lll} \Delta u=\displaystyle\frac{\lambda|x|^{\alpha}}{u^p} \ \ &\hbox{ in} \,\ \R ^2\setminus \{0\},\\[2mm] u(0)=0 \…

In this article we address the regularity of stable solutions to semilinear elliptic equations $-\Delta u = f(u)$ with MEMS type nonlinearities. More precisely, we will have $0\leq u \leq 1$ in a domain $\Omega \subset \mathbb{R}^n$ and…

We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } \Omega $$…

In this paper we study solutions to elliptic linear equations $L(u)=\partial_i(a^{ij}(x)\partial_j u) + b^i(x) \partial_i u + c(x) u=0$, either on $R^n$ or a Riemannian manifold, under the assumption of Lipschitz control on the coefficients…

We establish the existence of three solutions for singular semilinear elliptic system, two of which are of opposite constant-sign. Under a strong singularity effect, the third solution is nodal with synchronous sign components. The approach…

The aim of this paper is to study radial symmetry and monotonicity properties for positive solution of elliptic equations involving the fractional Laplacian. We first consider the semi-linear Dirichlet problem (-\Delta)^{\alpha} u=f(u)+g,\…

We consider the existence of \emph{normalized} solutions in $H^1(\R^N) \times H^1(\R^N)$ for systems of nonlinear Schr\"odinger equations which appear in models for binary mixtures of ultracold quantum gases. Making a solitary wave ansatz…

In this paper we prove existence results and asymptotic behavior for strong solutions $u\in W^{2,2}_{\textrm{loc}}(\Omega)$ of the nonlinear elliptic problem \begin{equation} \tag{P} \label{abstr} \left\{ \begin{array}{ll}…

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $\Omega$ in $\mathbb{R}^n$, $n \ge 2$: $$ -\sum_{i,j=1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f…

Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq1$) be a bounded and smooth domain and $a:\Omega\rightarrow\mathbb{R}$ be a sign-changing weight satisfying $\int_{\Omega}a<0$. We prove the existence of a positive solution $u_{q}$ for the problem…

Nodal solutions of a parametric (p_1,p_2)-Laplacian system, with Neumann boundary conditions, are obtained by chiefly constructing appropriate sub-super-solution pairs.

In this paper, we study a class of quasilinear Schr\"{o}dinger equation of the form $$-\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u &=&\lambda|u|^{q-2}u+|u|^{2^*(2\alpha)-2}u,\quad\mbox{in}{\mathbb{R}}^N,…

Using dual method we establish the existence of nodal ground state solution for the following class of problems $$ \left\{ \begin{array}{l} \Delta^2 u = f(u), \quad \mbox{in} \quad \Omega, \\ u =Bu=0,\quad\mbox{on} \quad \partial \Omega…