arXiv subjects
Publications and source records attributed to Claudianor O. Alves.
We show the existence of solution for some classes of nonlocal problems. Our proof combines the presence of sub and supersolution with the pseudomonotone operators theory.
The present paper concerns with the existence of blow-up solution for a class of elliptic system with convection term. Here, we prove a result involving sub and supersolution for a class of elliptic system whose nonlinearity can depend of the gradient of the solution. This result permits to study the existence of blow-up solution for a large class of systems.
The main goal this work is to prove two results like Strauss and Lions for Orlicz-Sobolev spaces. After, we use these results for study the existence of solutions for a class of quasilinear problems in $\mathbb{R}^{N}$.
In this paper we establish the existence of two positive solutions for the obstacle problem $$ \displaystyle \int_{\Re}\left[u'(v-u)'+(1+λV(x))u(v-u)\right] \geq \displaystyle \int_{\Re} f(u)(v-u), \forall v\in \Ka $$ where $f$ is a continuous function verifying some technical conditions and $\Ka$ is the convex set given by $$ \Ka =\left\{v\in H^{1}(\Re); v \geq φ\right\}, $$ with $φ\in H^{1}(\Re)$ 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.
We prove the existence of solutions for a class of quasilinear problems involving variable exponents and with nonlinearity having critical growth. The main tool used is the variational method, more precisely, Ekeland's Variational Principle and the Mountain Pass Theorem.
We prove the existence of solution for a class of $p(x)$-Laplacian equations where the nonlinearity has a critical growth. Here, we consider two cases: the first case involves the situation where the variable exponents are periodic functions. The second one involves the case where the variable exponents are nonperiodic perturbations.
We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines pseudomonotone operator theory, sub and supersolution method and perturbation arguments involving singular terms.
We prove the existence of least energy nodal solution for a class of Schrödinger-Poisson system in a bounded domain $Ω\subset \mathbb{R}^3$ with nonlinearity having a subcritical growth.
Using variational methods combined with perturbation arguments, we study the existence of nontrivial classical solution for the quasilinear Schrödinger equation \begin{equation*}\label{1.1} -Δu+ V(x)u+ \fracκ{2}[Δ|u|^2]u=l(u),\ x\in\mathbb{R}^N, \end{equation*} where $V:\mathbb{R}^N\rightarrow \mathbb{R}$ and $l:\mathbb{R} \to \mathbb{R}$ are continuous function, $κ$ is a parameter and $N\geq 3$. This model has been proposed in plasma physics and nonlinear optics. As a main novelty with respect to some previous results, we are able to deal with the case $κ>0$.
The multiplicity of positive weak solutions for a quasilinear Schrödinger equations $-L_p u +(λA(x)+1)|u|^{p-2}u= h(u)$ in $\mathbb{R}^N$ is established, where $L_p u\doteq ε^{p}Δ_p u +ε^{p}Δ_p (u^2)u$, $A$ is a nonnegative continuous function and nonlinear term $h$ has a subcritical growth. We achieved our results by using minimax methods and Lusternik-Schnirelman theory of critical points.
In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$ (-i\nabla - A(μx))^{2}u= μ|u|^{q-2}u + |u|^{2^{*}-2}u \ \mbox{in} \ Ω, \ \ \ \ u \in H^{1}_{0}(Ω,\mathbb{C}), $$ where $Ω\subset \mathbb{R}^{N} (N \geq 4)$ is a bounded domain with smooth boundary. Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of $Ω$.
We investigate the multiplicity of nontrivial weak solutions for a class of complex equations. This class of problems are related with the existence of solitary waves for a nonlinear Schödinger equation. The main result is established by using minimax methods and Lusternik-Schnirelman theory of critical points.