Existence and Multiplicity of Solutions for a Class of Elliptic Equations Without Ambrosetti-Rabinowitz Type Conditions

Autor(es)

Juarez Hurtado, E. ^[1] ; Miyagaki, O. H. ^[2] ; Rodrigues, R. S. ^[1]

Afiliação do(s) autor(es)

^[1] Univ Fed Sao Carlos, Dept Math, BR-13565905 Sao Carlos, SP - Brazil

^[2] Univ Fed Juiz de Fora, Dept Math, BR-36036330 Juiz De Fora, MG - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Journal of Dynamics and Differential Equations; v. 30, n. 2, p. 405-432, JUN 2018.
• Citações Web of Science: 1

Resumo

In this paper we establish, using variational methods, the existence and multiplicity of weak solutions for a general class of quasilinear problems involving -Laplace type operators, with Dirichlet boundary conditions involving variable exponents without Ambrosetti and Rabinowitz (A-R) type growth conditions, namely [- div(a(vertical bar del u vertical bar(p(x)))vertical bar del u vertical bar(p(x)-2)del vertical bar del u) - lambda f(x, u) in Omega u = 0 on partial derivative Omega. By different types of versions of the Mountain Pass Theorem with Cerami condition, as well as, the Fountain and Dual Theorem with Cerami condition, we obtain some existence of weak solutions for the above problem under some considerations. Moreover, we show that the problem treated has at least one nontrivial solution for any parameter small enough, and also that the solution blows up, in the Sobolev norm, as Finally, by imposing additional hypotheses on the nonlinearity we get the existence of infinitely many weak solutions by using the Genus Theory introduced by Krasnoselskii. (AU)

• Processo FAPESP: 15/11912-6 - Existência de solução para problemas elípticos
• Beneficiário: Rodrigo da Silva Rodrigues
• Modalidade de apoio: Auxílio à Pesquisa - Regular