相关论文: Existence of stable solutions to $(-\Delta)^m u=e^…

In this paper, we prove that all the positive solutions for the PDE system (-\Delta)^{k}u_{i} = f_{i}(u_{1},..., u_{m}), x \in R^{n}, i = 1, 2,..., m are super polyharmonic, i.e. (-\Delta)^{j}u_{i} > 0, j = 1, 2,..., k - 1; i = 1, 2,...,m.…

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[ \Delta^m u = \pm u^{\alpha} \quad \text{ in }…

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic…

In this paper, we present a result on the existence of ground state solutions for the polyharmonic nonlinear equation $(-\Delta)^m u=g(u)$, assuming that $g$ has a general subcritical growth at infinity, inspired by Berestycki and Lions…

We examine the elliptic system given by {equation} \label{system_abstract} -\Delta u = v^p, \qquad -\Delta v = u^\theta, \qquad \{in} \IR^N, {equation} for $ 1 < p \le \theta$ and the fourth order scalar equation {equation}…

We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-\Delta u = (1-u) u^m - \lambda u^n$ in a bounded domain $\Omega \subset…

Let $x_0\in\Omega\subset\Bbb{R}^n$, $n\ge 2$, be a domain and let $m\ge 2$. We will prove that a solution $u$ of the polyharmonic equation $\Delta^mu=0$ in $\Omega\setminus\{x_0\}$ has a removable singularity at $x_0$ if and only if…

This article studies (optimal) $W^{2m-1,\infty}$-regularity for the polyharmonic equation $(-\Delta)^m u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$, where $\Gamma$ is a (suitably regular) $(n-1)$-dimensional submanifold of $\mathbb{R}^n$,…

In this paper, we are devoted to studying the positive solutions of the following higher order Hardy-H\'enon equation $$ (-\Delta)^{m}u=|x|^{\alpha}u^{p} \quad\mbox{in}~ B_{1}\setminus\{0\}\subset\mathbb{R}^{n} $$ with an isolated…

We study the stability of radial solutions of the semilinear elliptic equation $\Delta u +f(u)=0$ in ${\bf R^N}$, where $N \geq 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with…

We study the existence of solution to the problem $$(-\Delta)^\frac n2u=Qe^{nu}\quad\text{in }\mathbb{R}^{n},\quad \kappa:=\int_{\mathbb{R}^{n}}Qe^{nu}dx<\infty,$$ where $Q\geq 0$, $\kappa\in (0,\infty)$ and $n\geq 3$. Using ODE techniques…

In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equations $$ \Delta_{p}^2 u =…

We examine the equation given by \begin{equation} \label{eq_abstract} -\Delta u + a(x) \cdot \nabla u = u^p \qquad \mbox{in $ \IR^N$,} \end{equation} where $p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We show…

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…

We classify the stable solutions (positive or sign-changing, radial or not) to the following nonlocal Lane-Emden equation: $(-\Delta)^s u=|u|^{p-1}u$ in $\mathbb{R}^n$ for $2<s<3$.

We prove existence results for the Lane-Emden type system \[ \begin{cases} \begin{aligned} (-\Delta)^{\alpha} u=\left| v \right|^q \\ (-\Delta)^{\beta} v= \left| u \right|^p \end{aligned} \text{ in } B_1 \subset \mathbb{R}^N \\…

In this paper we study stable solutions to the fractional equation \begin{align} (-\Delta)^s u =f(u), \quad |u| < 1 \quad \mbox{in $\mathbb{R}^d$}, \end{align}where $0<s<1$ and $f:[-1,1] \rightarrow \mathbb{R}$ is a $C^{1,\alpha}$ function…

This work is focused on the study of the nonlinear elliptic higher order equation \begin{equation}\nonumber \left( -\Delta \right)^m u = S_k[-u] + \lambda f, \qquad x \in \mathbb{R}^N, \end{equation} where the $k-$Hessian $S_k[u]$ is the…

We prove the existence of non-radial entire solution to $$\Delta^2 u+u^{-q}=0\quad\text{in }\mathbb{R}^3,\quad u>0,$$ for $q>1$. This answers an open question raised by P. J. McKenna and W. Reichel (E. J. D. E. \textbf{37} (2003) 1-13).