Related papers: Bubble towers for supercritical semilinear ellipti…

In this work we study the existence of nodal solutions for the problem $$ -\Delta u = \lambda u e^{u^2+|u|^p} \text{ in }\Omega, \; u = 0 \text{ on }\partial \Omega, $$ where $\Omega\subseteq \mathbb R^2$ is a bounded smooth domain and…

In this work we consider the boundary blow-up problem $$ \left\{ \begin{array}{ll} \Delta u = f(u) & \hbox{in } B\\ \ \ u=+\infty & \hbox{on }\partial B \end{array} \right. $$ where $B$ stands for the unit ball of $\mathbb{R}^N$ and $f$ is…

In this paper, the semilinear elliptic systems with Dirichlet boundary value are considered \begin{align} \left\{\begin{array}{ll} -\Delta v=f(u) & \mathrm{in}\ \Omega, -\Delta u=g(v) & \mathrm{in}\ \Omega, u=0, \ v=0 & \mathrm{on}\…

We establish quantitative asymptotic behaviors for nonnegative solutions of the critical semilinear equation $-\Delta u=u^{\frac{n+2}{n-2}}$ with isolated boundary singularities, where $n\ge 3$ is the dimension.

We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system \begin{equation}\label{e051} \left\{ \begin{split} &\Delta…

In this paper, we consider existence of positive solutions for the Schr\"odinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} \Delta_pu+\Delta_p(|u|^{2\gamma})|u|^{2\gamma-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\…

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation $$ -\Delta u + u = a(x)|u|^{p-2}u…

We consider the problem $\Delta u + \lambda u +u^5 = 0$, $u>0$, in a smooth bounded domain $\Omega$ in ${\mathbb R}^3$, under zero Dirichlet boundary conditions. We obtain solutions to this problem exhibiting multiple bubbling behavior at…

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…

Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive…

We consider the semilinear elliptic boundary value problem \[ -\Delta u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on }\partial\Omega, \] in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for supercritical…

We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a…

In this paper, we are concerned with the following elliptic equation $$ ( SC_\varepsilon ) \qquad \begin{cases} -\Delta u = |u|^{4/(n-2)}u [\ln (e+|u|)]^\varepsilon & \hbox{ in } \Omega,\\ u = 0 & \hbox{ on }\partial \Omega, \end{cases} $$…

If $p>1+2/n$ then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions…

In this paper, we mainly study the critical points and critical zero points of solutions $u$ to a kind of linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in a multiply connected domain $\Omega$ in $\mathbb{R}^2$.…

We prove the monotonicity of positive solutions to the problem $-\Delta u = f(u)$ in $\mathbb{R}^N_+ := \{(x',x_N)\in\mathbb{R}^N \mid x_N>0 \}$ under zero Dirichlet boundary condition with a possible singular nonlinearity $f$. In some…

We deals with nonlinear elliptic Dirichlet problems of the form $${\rm div}(|D u|^{p-2}D u )+f(u)=0\quad\mbox{ in }\Omega,\qquad u\in H^{1,p}_0(\Omega) $$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\ge 2$, $p> 1$ and $f$ has…

We show that in some suitable torus-like domains D some supercritical elliptic problems have an arbitrary large number of sign-changing solutions with alternate positive and negative layers which concentrate at different rates along a…

In this paper, we prove the existence of nontrivial unbounded domains $\Omega\subset\mathbb{R}^{n+1},n\geq1$, bifurcating from the straight cylinder $B\times\mathbb{R}$ (where $B$ is the unit ball of $\mathbb{R}^n$), such that the…

We consider the energy critical heat equation in $\mathbb R^n$ for $n\ge 7$ $$\left\{ \begin{aligned} u_t & = \Delta u+ |u|^{\frac 4{n-2}}u \hbox{ in }\ \mathbb R^n \times (0, \infty), \\ u(\cdot,0) & = u_0 \ \hbox{ in }\ \mathbb R^n,…