Related papers: Remarks on two fourth order elliptic problems in w…

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…

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem \[ \begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases} \] that branch…

In this paper, we primarily consider the following semilinear elliptic equation \begin{eqnarray*} \arraycolsep=1pt\left\{ \begin{array}{lll} \displaystyle -\Delta u= h(x,u)\quad \ &{\rm in}\ \Omega,\\[1.5mm] \phantom{ -\Delta }…

We prove that the equation \begin{eqnarray*} -\Delta_p u =\lambda\Big( \frac{1} {u^\delta} + u^q + f(u)\Big)\;\text{ in } \, B_R(0) u =0 \,\text{ on} \; \partial B_R(0), \quad u>0 \text{ in } \, B_R(0) \end{eqnarray*} admits a weak radially…

We investigate here the degenerate bi-harmonic equation: $$\Delta_{m}^2 u=f(x,u)\; \;\;\mbox{in} \O,\quad u = \Delta u = 0\quad \mbox{on }\; \p\Omega,$$ with $m\ge 2,$ and also the degenerate tri-harmonic equation: $$ -\Delta_{m}^3…

In this paper, we study the exterior Dirichlet problem for the fully nonlinear elliptic equation $f(\lambda(D^{2}u))=1$. We obtain the necessary and sufficient conditions of existence of radial solutions with prescribed asymptotic behavior…

In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-\Delta)^\alpha u=u^p\quad &{\rm in}\quad…

Starting from a bound state (positive or sign-changing) solution to $$ -\Delta \omega_m =|\omega_m|^{p-1} \omega_m -\omega_m \ \ \mbox{in}\ \R^n, \ \omega_m \in H^2 (\R^n)$$ and solutions to the Helmholtz equation $$ \Delta u_0 + \lambda…

We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of {equation} \label{pop} -div(\omega_1 \nabla u) = \omega_2 f(u) \qquad \text{in}\ \ \IR^N, {equation} with positive smooth weights $…

In this paper we study a class of fractional elliptic problems of the form $$ \Ds u= f(x,u) \quad \textrm{in} \O u=0\quad \textrm{in} \R^N \setminus \O,$$ where $s\in(0,1)$. We prove nonexistence of positive solutions when $\O$ is…

Let $0<\alpha<2$ be any real number. In this paper, we investigate the following semilinear equations involving the fractional Laplacian \begin{equation}(-\bigtriangleup)^{\alpha/2} u(x)=f(u),\end{equation} on $\mathbb{R}^n$ and…

We show non-existence of solutions of the Cauchy problem in $\mathbb{R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial_t u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities $\phi$ .…

We study the semilinear elliptic equation \begin{equation*} -\Delta u=u^\alpha |\log u|^\beta\quad\text{in }B_1\setminus\{0\}, \end{equation*} where $B_1\subset\mathbb{R}^n$ with $n\geq 3$, $\frac{n}{n-2} < \alpha < \frac{n+2}{n-2}$ and…

Let us consider a semilinear boundary value problem $ - \Delta u= f(x,u),$ in $\Omega,$ with Dirichlet boundary conditions, where $ \Omega \subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions…

We study the semilinear elliptic problem \[ -\Delta u = Q_{\Omega} |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where \( Q_{\Omega} = \chi_{\Omega} - \chi_{\mathbb{R}^N \setminus \Omega} \) for a bounded smooth domain \( \Omega \subset…

We study existence and stability properties of entire solutions of a polyharmonic equation with an exponential nonlinearity. We study existence of radial entire solutions and we provide some asymptotic estimates on their behavior at…

The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind.…

We study the existence of solutions to the problem $$ (-\Delta)^{\frac{n}{2}}u = Qe^{nu}\quad\text{in }\mathbb{R}^n, \quad V := \int_{\mathbb{R}^n}e^{nu}dx < \infty,$$ where $Q=(n-1)!$ or $Q=-(n-1)!$. Extending the works of Wei-Ye and…

In this work we prove the existence of infinitely many nonradial solutions that change signal to the problem $-\Delta u=f(u)$ in $B$ with $u=0$ on $\partial B$, where $B$ is the unit ball in $\mathbb{R}^2$ and $f$ is a continuous and odd…

We consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction…