A nonlocal logistic equation with nonlinear advection term
In this paper, we study a nonlocal logistic equation with nonlinear advection term.
math.AP↗
arXiv subjects
Romildo N. de Lima
Publications and source records attributed to Romildo N. de Lima.
On an Ambrosetti-Prodi type problem in $\R^N$
In this paper we study results of existence and non-existence of solutions for the following Ambrosetti-Prodi type problem $$ \left\{ \begin{array}{lcl} -Δu=P(x)\Big( g(u)+f(x)\Big) \mbox{ in } \mathbb{R}^N,\\ u \in D^{1,2}(\R^N),\ \lim_{|x|\to +\infty}u(x)=0, \end{array} \right. \eqno{(P)} $$ where $N\geq3$, $P\in C(\R^N,\R^+)$, $f\in C(\R^N)\cap L^{\infty}(\R^N)$ and $g\in C^1(\R)$. The main tools used are the sub-supersolution method and Leray-Schauder topological degree theory.
math.AP↗
Existence of positive solution for a system of elliptic equations via bifurcation theory
In this paper we study the existence of solution for the following class of system of elliptic equations $$ \left\{ \begin{array}{lcl} -Δu=\left(a-\int_ΩK(x,y)f(u,v)dy\right)u+bv,\quad \mbox{in} \quad Ω -Δv=\left(d-\int_ΩΓ(x,y)g(u,v)dy\right)v+cu,\quad \mbox{in} \quad Ωu=v=0,\quad \mbox{on} \quad \partialΩ \end{array} \right. \eqno{(P)} $$ where $Ω\subset\R^N$ is a smooth bounded domain, $N\geq1$, and $K,Γ:Ω\timesΩ\rightarrow\R$ is a nonnegative function checking some hypotheses and $a,b,c,d\in\R$. The functions $f$ and $g$ satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for $(P)$.
math.AP↗
Existence of solution for a nonlocal problem in $\R^N$ via bifurcation theory
In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -Δu=\left(λf(x)-\int_{\R^N}K(x,y)|u(y)|^γdy\right)u,\quad \mbox{in} \quad \R^{N}, \\ \displaystyle \lim_{|x| \to +\infty}u(x)=0,\quad u>0 \quad \text{in} \quad \R^{N}, \end{array} \right. \eqno{(P)} $$ where $N\geq3$, $λ>0, γ\in[1,2)$, $f:\R\rightarrow\R$ is a positive continuous function and $K:\R^N\times\R^N\rightarrow\R$ is a nonnegative function. The functions $f$ and $K$ satisfy some conditions, which permit to use Bifurcation Theory to prove the existence of solution for problem $(P)$.
math.AP↗
Choquard Equations with Mixed Potential
In this paper, we study the following class of nonlinear Choquard equation, $$-Δu+a(z)u=K(u)f(u)\quad \text{in}\quad \R^N,$$ where $\R^N=\R^L\times\R^M$, $L\geq2$, $K(u)=|.|^{-γ}*F(u)$, $γ\in(0,N)$, $a$ is a continuous real function and $F$ is the primitive function of $f$. Under some suitable assumptions mixed on the potential $a$. We prove existence of a nontrivial solution for the above equation.
math.AP↗