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Superlinear problems without Ambrosetti and Rabinowitz growth condition
(Journal of Differential Equations, 2008-12-15) Miyagaki, O. H.; Souto, M. A. S.
Superlinear elliptic boundary value problems without Ambrosetti and Rabinowitz growth condition are considered. Existence of nontrivial solution result is established by combining some arguments used by Struwe and Tarantello and Schechter and Zou (also by Wang and Wei). Firstly, by using the mountain pass theorem due to Ambrosetti and Rabinowitz is constructed a solution for almost every parameter λ by varying the parameter λ. Then, it is considered the continuation of the solutions.
Subcritical perturbations of a singular quasilinear elliptic equation involving the critical Hardy–Sobolev exponent
(Nonlinear Analysis: Theory, Methods & Applications, 2007-03-15) Miyagaki, O. H.; Assunção, R. B.; Carrião, P. C.
In this work we improve some known results for a singular operator and also for a wide class of lower-order terms by proving a multiplicity result. The proof is made by applying the generalized mountain-pass theorem due to Ambrosetti and Rabinowitz. To do this, we show that the minimax levels are in a convenient range by combining a special class of approximating functions, due to Gazzola and Ruf, with the concentrating functions of the best Sobolev constant.
On positive solutions for a class of singular quasilinear elliptic systems
(Journal of Mathematical Analysis and Applications, 2007-10-15) Miyagaki, O. H.; Rodrigues, R. S.
We study through the lower and upper-solution method, the existence of positive weak solution to the quasilinear elliptic system with weights ⎧ ⎪ −div(|x|−ap |∇u|p−2 ∇u) = λ|x|−(a+1)p+c1 uα v γ in Ω, ⎨ −div(|x|−bq |∇v|q−2 ∇v) = λ|x|−(b+1)q+c2 uδ v β in Ω, ⎪ ⎩ u=v=0 on ∂Ω, −p −q where Ω is a bounded smooth domain of RN , with 0 ∈ Ω, 1 < p, q < N , 0 a < N p , 0 b < N q , 0 α < p − 1, 0 β < q − 1, δ, γ , c1 , c2 > 0 and θ := (p − 1 − α)(q − 1 − β) − γ δ > 0, for each λ > 0.
Multiple positive solutions for semilinear Dirichlet problems with sign-changing weight function in infinite strip domains
(Nonlinear Analysis: Theory, Methods & Applications, 2009-10-01) Miyagaki, O. H.; Miotto, M. L.
In this paper, existence and multiplicity results to the following Dirichlet problem −∆u + u = λf (x)|u|q−1 + h(x)|u|p−1 , u > 0, u = 0, in Ω in Ω on ∂ Ω are established, where Ω = Ω × R, Ω ⊂ RN −1 is bounded smooth domain and N ≥ 2. Here 1 < q < 2 < p < 2∗ 2∗ = N2N2 if N ≥ 3, 2∗ = ∞ if N = 2 , λ is a positive real − parameter, the function f , among other conditions, can possibly change sign in Ω , and the function h satisfies suitable conditions. The study is based on the comparison of energy levels on Nehari manifold.
On positive solution for a class of degenerate quasilinear elliptic positone/semipositone systems
(Nonlinear Analysis: Theory, Methods & Applications, 2009-01-01) Miyagaki, O. H.; Rodrigues, R. S.
This paper deals with the existence and nonexistence of positive weak solutions of degenerate quasilinear elliptic systems with subcritical and critical exponents. The nonlinearities involved have semipositone and positone structures and the existence results are obtained by applying the lower and upper-solution method and variational techniques.
Non-autonomous perturbations for a class of quasilinear elliptic equations on R
(Journal of Mathematical Analysis and Applications, 2008-08-01) Miyagaki, O. H.; Alves, M. J.; Carrião, P. C.
This paper is concerned with the existence of two positive solutions for a class of quasilinear elliptic equations on R involving the p-Laplacian, with a non-autonomous perturbation. The first solution is obtained as a local minimum in a neighborhood of 0 and the second one by a mountain-pass argument. The special features of the problem here is the “complex” structure of the linear part which, in particular, oblige to work into the space W 1,p (R). Then one faces problems in the convergence of the sequences of derivatives un → u.
Critical singular problems on unbounded domains
(Abstract and Applied Analysis, 2004-04-30) Morais Filho, D. C. de; Miyagaki, O. H.
We present some results of existence for the following problem: − ∆u = a(x)g(u)+u | u | 2 − 2 , x ∈ R N (N ≥ 3), u ∈ D 1,2 ( R N ), where the function a is a sign-changing function with a singularity at the origin and g has growth up to the Sobolev critical exponent 2 ∗ = 2N/(N − 2).

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