On a class of mixed Choquard-Schrödinger-Poisson systems
Marius Ghergu Gurpreet Singh
Discrete & Continuous Dynamical Systems - S 2019, 12(2): 297-309 doi: 10.3934/dcdss.2019021
We study the system
$\left\{\begin{split}-\Delta u+u&=(I_\alpha*|u|^p)|u|^{p-2}u+K(x) \phi |u|^{q-2}u & \qquad \mbox{ in }\mathbb{R}^N,\\-\Delta \phi&=K(x)|u|^q& \qquad \mbox{ in }\mathbb{R}^N,\end{split}\right.$
$N≥ 3$
$α∈ (0,N)$
$K≥ 0$
. Using a Pohozaev type identity we first derive conditions in terms of
for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.
keywords: Choquard equation Schrödinger-Poisson equation Pohozaev identity ground state solution
Nonradial blow-up solutions of sublinear elliptic equations with gradient term
Marius Ghergu Vicenţiu Rădulescu
Communications on Pure & Applied Analysis 2004, 3(3): 465-474 doi: 10.3934/cpaa.2004.3.465
Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, su$p_{s\geq 1} f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation $\Delta u+|\nabla u|=p(x)f(u)$ in $\Omega,$ where $\Omega$ is either a smooth bounded domain or $\Omega=\mathbb R^N$. If $\Omega$ is bounded we prove that the above problem has never a blow-up boundary solution. Since $f$ does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case $\Omega=\mathbb R^N$ a necessary and sufficient condition for the existence of a positive solution that blows up at infinity.
keywords: maximum principle Explosive solution elliptic equation sublinear growth condition.
A new critical curve for the Lane-Emden system
Wenjing Chen Louis Dupaigne Marius Ghergu
Discrete & Continuous Dynamical Systems - A 2014, 34(6): 2469-2479 doi: 10.3934/dcds.2014.34.2469
We study stable positive radially symmetric solutions for the Lane-Emden system $-\Delta u=v^p$ in $\mathbb{R}^N$, $-\Delta v=u^q$ in $\mathbb{R}^N$, where $p,q\geq 1$. We obtain a new critical curve that optimally describes the existence of such solutions.
keywords: singular solutions. stable solutions critical curve radially symmetric solutions Lane-Emden system
Professor Vicenţiu Rǎdulescu celebrates his sixtieth anniversary
Hugo Beirão da Veiga Marius Ghergu Alberto Valli
Discrete & Continuous Dynamical Systems - S 2019, 12(2): ⅰ-ⅳ doi: 10.3934/dcdss.201902i

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