scispace - formally typeset
Search or ask a question
Author

Elard Juarez Hurtado

Bio: Elard Juarez Hurtado is an academic researcher from Federal University of São Carlos. The author has contributed to research in topics: Mathematics & Dirichlet boundary condition. The author has an hindex of 2, co-authored 4 publications receiving 11 citations.

Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, the authors show that the problem has at least one solution, which converges to zero, in the norm of the space X as a function of the variable exponent spaces with critical growth.
Abstract: In this paper, we establish existence and asymptotic behaviour of nontrivial weak solution of a class of quasilinear stationary Kirchhoff type equations involving the variable exponent spaces with critical growth, namely $$\begin{aligned}{\left\{ \begin{array}{ll} -M (\mathcal{A}(u)) {\rm div} (a(| abla u|^{p(x)}) | abla u|^{p(x) - 2} abla u) = \lambda f (x, u) + |u|^{s(x)-2} u \quad {\rm in} \quad \Omega,\\ u = 0 \quad {\rm on} \quad \partial \Omega,\end{array}\right. } \end{aligned}$$ where $${\Omega}$$ is a bounded smooth domain of $${\mathbb{R}^N}$$ , with homogeneous Dirichlet boundary conditions on $${\partial \Omega}$$ , the nonlinearities $${f : \Omega \times \mathbb{R} \rightarrow \mathbb{R}}$$ is a continuous function, $${a : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}}$$ is a function of the class $${C^1}$$ , $${M : \mathbb{R}^{+}_{0} \rightarrow \mathbb{R}^{+}}$$ is a continuous function whose properties will be introduced later, and $${\lambda}$$ is a positive parameter. We assume that $${\mathcal{C} = \{x \in \Omega : s(x) = \gamma^{*}(x)\} eq \emptyset}$$ , where $${\gamma (x)^{*} = N \gamma (x) / (N - \gamma (x))}$$ is the critical Sobolev exponent. We show that the problem has at least one solution, which it converges to zero, in the norm of the space X as $${\lambda \rightarrow + \infty}$$ . Our result extends, complement and complete in several ways some of the recent works. We want to emphasize that a difference of some previous research is that the conditions on $${a(\cdot)}$$ are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for $${p(x) > 1}$$ , for all $${x \in \bar{\Omega}}$$ . The main tools used are the Mountain Pass Theorem without the Palais-Smale condition given in [11] and the Concentration Compactness Principle for variable exponent found in [9]. We remark that it will be necessary a suitable truncation argument in the Euler- Lagrange operator associated.

16 citations

Posted Content
TL;DR: In this article, the authors studied the existence and multiplicity of weak solutions for a general class of elliptic equations in a smooth bounded domain, driven by a nonlocal integrodifferential operator.
Abstract: In this paper, we study the existence and multiplicity of weak solutions for a general class of elliptic equations (\mathscr{P}_{\lambda}) in a smooth bounded domain, driven by a nonlocal integrodifferential operator \mathscr{L}_{\mathcal{A}K} with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz type growth conditions. Using different versions of the Mountain Pass Theorem, as well as, the Fountain Theorem and Dual Fountain Theorem with Cerami condition, we obtain the existence of weak solutions for the problem (\mathscr{P}_{\lambda}) and we show that the problem treated has at least one nontrivial solution for any parameter \lambda>0 small enough as well as that the solution blows up, in the fractional Sobolev norm, as \lambda \to 0. Moreover, for the case sublinear, by imposing some additional hypotheses on the nonlinearity f(x,\cdot), by using a new version of the symmetric Mountain Pass Theorem due to Kajikiya [36], we obtain the existence of infinitely many weak solutions which tend to be zero, in the fractional Sobolev norm, for any parameter \lambda>0. As far as we know, the results of this paper are new in the literature.

6 citations

Posted Content
TL;DR: In this paper, the authors studied the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $\wp(\cdot)-$Laplacian operator involving constant/variable exponent.
Abstract: In this manuscript, we will study the asymptotic behavior for a class of nonlocal diffusion equations associated with the weighted fractional $\wp(\cdot)-$Laplacian operator involving constant/variable exponent. In the case of constant exponents, under some appropriate conditions, we will study the existence of solutions and asymptotic behavior of solutions by employing the subdifferential approach and we will study the problem when $\wp$ goes to $\infty$. Already, for case the weighted fractional $\wp(\cdot)$-Laplacian operator, we will also study the asymptotic behavior of the problem solution when $\wp(\cdot)$ goes to $\infty$, in the whole or in a subset of the domain (the problem involving the fractional $\wp(\cdot)$-Laplacian presents a discontinuous exponent). To obtain the results of the asymptotic behavior in both problems it will be via Mosco convergence.

1 citations


Cited by
More filters
Dissertation
13 May 2014
TL;DR: In this paper, the authors studied the existence, uniqueness, comparison results and asymptotic behavior of the solutions of some nonlocal diffusion problems, which are set in metric measure spaces.
Abstract: This thesis studies the existence, uniqueness, comparison results and asymptotic behaviour of the solutions of some nonlocal di¬ffusion problems. All the problems in this thesis are set in metric measure spaces, which are introduced in Chapter 1. These spaces include very different type of spaces, for example, open subsets in R^N, graphs, manifolds, multistructures or some fractal sets. In Chapter 2, we study basic properties of the nonlocal diffusion ope¬rador which will be used in the following chapters. In particular, we study regularity, compactness, positiveness and the spectrum of the operator. In Chapter 3, we study the solutions of the linear nonlocal diffusion problems. In particular we describe the asymptotic beha¬viour using spectral methods. In Chapter 4, we study the nonlinear nonlocal diffusion problem with a local reaction. In particular, we prove weak and strong maximum principles and the existence of two extremal equilibria, which attract the asymptotic dynamics of the solutions. We also show how the lack of smoothing prevent us from proving the existence of a global attractor. In Chapter 5, we consider a nonlinear term that is a nonlinear function of the ave¬rage of the solution in a ball. In this case we prove that there are strong restrictions for the weak and strong maximum principles to hold. When these hold, we prove the existence of a global attrac¬tor. In Chapter 6, we study sign-changing solutions of the nonlocal two-phase Stefan problem.Esta tesis estudia la existencia, unicidad, resultados de comparacion y comportamiento asintotico de las soluciones de algunos problemas de difusion no local. Todos los problemas en esta tesis estan considerados en espacios metricos de medida, que son introducidos en el Capitulo 1. Estos espacios incluyen muchos tipos diferentes de espacios, por ejemplo conjuntos abiertos en R^N, grafos, variedades, multiestructuras o algunos conjuntos fractales. En el Capitulo 2, estudiamos propiedades basicas de operadores de difusion no local que se utilizaran en los siguientes capitulos. En particular, estudiamos la regularidad, compacidad, positividad y el espectro del operador. En el Capitulo 3, estudiamos las soluciones de los problemas lineales de difusion no local. En particular, describimos el comportamiento asintotico usando metodos espectrales. En el Capitulo 4, estudiamos los problemas no lineales de difusion no local con reaccion local. En particular probamos principios debil y fuerte del maximo y la existencia de dos equilibrios maximales, que atraen la dinamica asintotica de las soluciones. Tambien mostramos la falta de regularizacion lo que no nos permite probar la existencia de un atractor global. En el Capitulo 5, consideramos un termino no lineal que es una funcion no lineal del promedio de la solucion en una bola. En este caso probamos que existen restricciones fuertes para que se cumplan los principios debil y fuerte del maximo. Cuando estas se cumplen, probamos la existencia de un atractor global. En el Capitulo 6, estudiamos soluciones que cambian de signo del problema no local de Stefan de dos fases.

110 citations

Journal ArticleDOI
TL;DR: In this article , the authors considered a class of quasilinear stationary Kirchhoff type potential systems with Neumann boundary conditions and established existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais-Smale condition.
Abstract: In this paper, we consider a class of quasilinear stationary Kirchhoff type potential systems with Neumann Boundary conditions, which involves a general variable exponent elliptic operator with critical growth. Under some suitable conditions on the nonlinearities, we establish existence and multiplicity of solutions for the problem by using the concentration-compactness principle of Lions for variable exponents and the mountain pass theorem without the Palais–Smale condition.

22 citations

Journal ArticleDOI
TL;DR: In this paper, a non-local version of the divergence theorem for fractional p(.,.)-Laplacian Laplacians with sign-changing potentials was established.
Abstract: In this paper we study the fractional p(., .)-Laplacian and we introduce the corresponding nonlocal conormal derivative for this operator. We prove basic properties of the corresponding function space and we establish a nonlocal version of the divergence theorem for such operators. In the second part of this paper, we prove the existence of weak solutions of corresponding p(., .)-Robin boundary problems with sign-changing potentials by applying variational tools.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the existence of nonegative solutions to a class of nonlinear boundary value problems of the Kirchhoff type was proved when the problem has discontinuous nonlinearity and critical Caffarelli-Kohn-Nirenberg growth.
Abstract: In this paper, we study the existence of nonegative solutions to a class of nonlinear boundary value problems of the Kirchhoff type. We prove existence results when the problem has discontinuous nonlinearity and critical Caffarelli–Kohn–Nirenberg growth.

8 citations

Journal ArticleDOI
TL;DR: In this paper, the authors established a concentration-compactness principle for the Sobolev space W2,p(⋅)(Ω)∩W01, p(⊆) ∩W1,p ∆)
Abstract: We establish a concentration-compactness principle for the Sobolev space W2,p(⋅)(Ω)∩W01,p(⋅)(Ω) that is a tool for overcoming the lack of compactness of the critical Sobolev imbedding. Using this r...

7 citations