Related papers: Stable and singular solutions of the equation $\De…

Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\Delta u + n (n - 2) u^{{n + 2}\over {n - 2}} = 0$ in $R^n (n \ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the…

We study the behaviour near a boundary point a of any positive solution of a nonlinear elliptic equations with forcing term which vanishes on the boundary except at a. Our results are based upon a priori estimates for solutions and…

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 extend the interior regularity results for stable solutions in [Cabr\'{e}, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear…

We consider the nonlinear elliptic equation \begin{equation*} -\Delta u + V(x)u = f(u), \qquad u\in D^{1,2}_0(\Omega), \end{equation*} in an exterior domain $\Omega$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at…

We consider the supercritical elliptic problem -\Delta u = \lambda e^u, \lambda > 0, in an exterior domain $\Omega = \mathbb{R}^N \setminus D$ under zero Dirichlet condition, where D is smooth and bounded in \mathbb{R}^N, N greater or equal…

Let $2\le n\le9$. Suppose that $f:R\to R$ is locally Lipschitz function satisfying $f(t)\ge A\min\{0,t\}-K$ for all $t\in R$ with some constant $A\ge0$ and $K\ge 0$. We establish an a priori interior H\"older regularity of $C^2$-stable…

We study positive solutions of equation (E) $-\Delta u + u^p|\nabla u|^q= 0$ ($0<p$, $0\leq q\leq 2$, $p+q>1$) and other related equations in a smooth bounded domain $\Omega \subset {\mathbb R}^N$. We show that if $N(p+q-1)<p+1$ then, for…

Given $B_1(0)$ the unit ball of $\mathbb{R}^n$ ($n\geq 3$), we study smooth positive singular solutions $u\in C^2(B_1(0)\setminus \{0\})$ to $-\Delta u=\frac{u^{2^\star(s)-1}}{|x|^s}-\mu u^q$. Here $0< s<2$, $2^\star(s):=2(n-s)/(n-2)$ is…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq1$, let $K$, $M$ be two nonnegative functions and let $\alpha,\gamma>0$. We study existence and nonexistence of positive solutions for singular problems of the form $-\Delta…

Several results about positive solutions -in a Lipschitz domain- of a nonlinear elliptic equation in a general form $ \Delta u(x)-g(x,u(x))=0$ are proved, extending thus some known facts in the case of $ g(x,t)=t^q$, $q>1$, and a smooth…

We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schr\"{o}dinger equations $$ iu_t+\Delta u=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}^2\times\mathbb{R}$,}$$ in the energy class.

In this paper we establish uniqueness criteria for positive radially symmetric finite energy solutions of semilinear elliptic systems of the form \begin{align*} \begin{aligned} - \Delta u &= f(|x|,u,v)\quad\text{in}\R^n, - \Delta v &=…

We are interested in the following semilinear elliptic problem: \begin{equation*} \begin{cases} -\Delta u + \lambda u = u^{p-1} \ \text{in} \ T,\\ u > 0, u = 0 \ \text{on} \ \partial T,\\ \int_{T}u^{2} \, dx= c \end{cases} \end{equation*}…

In this paper, we study asymptotic behavior of solution near 0 for a class of elliptic problem. The uniqueness of singular solution is established

We study the solvability in the whole Euclidean space of coercive quasi-linear and fully nonlinear elliptic equations modeled on $\Delta u\pm g(|\nabla u|)= f(u)$, $u\ge0$, where $f$ and $g$ are increasing continuous functions. We give…

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: \Delta u=u v^2 in R^n, \Delta v= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove…

We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…

We study radial symmetry of large solutions of the semi-linear elliptic problem \Delta u + \nabla h.\nabla u = f(|x|,u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of…

In this work, we study the of positive ground state solution for the semilinear elliptic problem $$ \left\{ \begin{array} [c]{ll}% -\Delta u=u^{p(x)-1},\quad u>0 & \mathrm{in}\,G\subseteq\mathbb{R}^{N}% ,\,N\geq3\\ u\in D_{0}^{1,2}(G), &…