An existence result for pp-Laplace equation with gradient nonlinearity in RN\mathbb{R}^N

We prove the existence of a weak solution to the problem \begin{equation*} \begin{split} -\Delta_{p}u+V(x)|u|^{p-2}u & =f(u,|\nabla u|^{p-2}\nabla u), \ \ \ \\ u(x) & >0\ \ \forall x\in\mathbb{R}^{N}, \end{split} \end{equation*} where $\Delta_{p}u=\hbox{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and the nonlinearity $f:\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is continuous and it depends on gradient of the solution. We use an iterative technique based on the Mountain pass theorem to prove our existence result.Comment: 10 pages, 0 figure

Positive solutions for nonvariational Robin problems

We study a nonlinear Robin problem driven by the $p$-Laplacian and with a reaction term depending on the gradient (the convection term). Using the theory of nonlinear operators of monotone-type and the asymptotic analysis of a suitable perturbation of the original equation, we show the existence of a positive smooth solution

A landesman-lazer local condition for nonlinear elliptic problems

Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2017.Texto parcialmente liberado pelo autor. Conteúdo restrito: Capítulos 1 e 2.O objetivo deste trabalho é estudar a existência, multiplicidade e não existência de soluções para problemas elípticos não-lineares dependendo de um parâmetro sob uma hipótese do tipo Landesman-Lazer. Para estabelecer a existência de solução combinamos o Método de Redução de Lyapunov-Schmidt e a técnica de congelamento do termo gradiente com argumentos de truncamento e aproximação através de métodos de bootstrap. Não há restrição de crescimento no infinito sobre o termo não-linear o qual pode mudar de sinal.CNPqThe purpose of this work is to study existence, multiplicity and non existence of solutions for nonlinear elliptic problems depending on a parameter under Landesman-Lazer type hypotheses. In ordem to establish the existence of solution we combine the Lyapunov-Schmidt Reduction Method and the term gradient freeze technique with truncation and approximation arguments via bootstrap methods. There is no growth restriction at infinity on the nonlinear term and it may change sign

Multiplicity of Positive Solutions for an Obstacle Problem in R

In this paper we establish the existence of two positive solutions for the obstacle problem \displaystyle \int_{\Re}\left[u'(v-u)'+(1+\lambda V(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka where $f$ is a continuous function verifying some technical conditions and \Ka is the convex set given by \Ka =\left\{v\in H^{1}(\Re); v \geq \varphi \right\}, with $\varphi \in H^{1}(\Re)$ having nontrivial positive part with compact support in $\Re$. \vspace{0.2cm} \noindent \emph{2000 Mathematics Subject Classification} : 34B18, 35A15, 46E39. \noindent \emph{Key words}: Obstacle problem, Variational methods, Positive solutions.Comment: To appear in Progress in Nonlinear Differential Equations and their Application