Author
Olímpio H. Miyagaki
Other affiliations: Universidade Federal de Viçosa
Bio: Olímpio H. Miyagaki is an academic researcher from Universidade Federal de Juiz de Fora. The author has contributed to research in topics: Sobolev space & Mountain pass theorem. The author has an hindex of 24, co-authored 142 publications receiving 2129 citations. Previous affiliations of Olímpio H. Miyagaki include Universidade Federal de Viçosa.
Papers published on a yearly basis
Papers
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TL;DR: In this article, the authors studied the solvability of problems of the type "updelta u = f(x,u), u = 0, u = √ √ u √ 0, √ U √ 1, √ n √ N √ O(n, n) where n is some bounded domain in R 2 and n is the maximal growth on n.
Abstract: In this paper we study the solvability of problems of the type \( - \Updelta u = f(x,u)\;{\text{in}}\;\Upomega ,\;u = 0\;{\text{on}}\;\partial \Upomega , \) where \( \Upomega \) is some bounded domain in R 2, and the function f(x, s) has the maximal growth on s which allows to treat problem variationally in \( H_{0}^{1} (\Upomega ) \).
327 citations
TL;DR: In this article, the authors considered superlinear elliptic boundary value problems without Ambrosetti and Rabinowitz growth condition and established the existence of nontrivial solution result by combining some arguments used by Struwe and Tarantello and Schechter and Zou (also by Wang and Wei).
199 citations
TL;DR: In this paper, the authors established the existence of standing wave solutions for quasilinear Schrodinger equations involving critical growth by using a change of variables, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem.
156 citations
94 citations
TL;DR: In this article, the authors considered quasilinear elliptic equations in R 2 of second order with critical exponential growth and obtained a Cerami sequence converging weakly to a solution v. In the proof that v is nontrivial, the main tool is the concentration compactness principle.
Abstract: Quasilinear elliptic equations in R 2 of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H 1 ( R 2 ) and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v . In the proof that v is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincare Anal. Non. Lineaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality.
91 citations
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TL;DR: In this paper, the existence of nontrivial solutions to the following problem was studied: {u∈W1,N(ℝN),u≥0, 0, and−div(|∇u|N−2 ∇u)
Abstract: We study the existence of nontrivial solutions to the following problem: {u∈W1,N(ℝN),u≥0 and−div(|∇u|N−2∇u)
257 citations
TL;DR: In this article, the existence of homoclinic orbits for the second order Hamiltonian system q ¨ + V q ( t, q ) = f ( t ), where q ∈ R n and V ∈ C 1 ( R × R n, R ), V ( t, q ) is T-periodic in t.
238 citations
TL;DR: In this article, the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p ( x ) -polyharmonic Kirchhoff equations, including the new delicate degenerate case, was established.
Abstract: In this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p ( x ) -polyharmonic Kirchhoff equations, including the new delicate degenerate case, not yet covered in the literature. The main tool is the symmetric mountain pass theorem of Ambrosetti and Rabinowitz.
205 citations
TL;DR: In this paper, an interpolation of Hardy inequality and Trudinger-Moser inequality in R N (N ≥ 2) was established, which can be used to establish if and only if is the volume of the unit sphere sufficient conditions under which the quasilinear nonhomogeneous partial differential equation has weak solutions.
Abstract: We establish an interpolation of Hardy inequality and Trudinger–Moser inequality in R N (N ≥ 2). Denote u u
1,τ ≤1 1,τ RN = R N (|∇u| 1 |x|β eα|u| N+ τ |u| N )dx N/(N−1)N−2−m=0 1/N for any τ > 0. There holds
α m|u|mN/(N−1) m!dx < ∞ 1/(N−1)βα, ω N−1 α N + N ≤ 1, where 0 ≤ β < N, α N = Nω N−1 N−1 . The above interpolation can be used to establish if and only if is the volume of the
unit sphere sufficient conditions under which the quasilinear nonhomogeneous partial differential equation
−Nu+V(x)|u| N−2 u =f(x,u)|x|β + h(x) in R N
has weak solutions, where −uous potential, f behaves like
N−2 ∇u) is the N-Laplacian, V isN u = −div(|∇u|N/(N−1)
when |u| → ∞, h ∈ (W1,N (R N ))∗ , 0 < β eα|u|
> 0.
173 citations
TL;DR: In this paper, the authors established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations with critical growth in the plane.
Abstract: It is established the existence of solutions for a class of asymptotically periodic quasilinear elliptic equations in $${\mathbb{R}^N}$$
with critical growth. Applying a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in $${H^1(\mathbb{R}^N)}$$
and satisfy the geometric hypotheses of the Mountain Pass Theorem. The Concentration–Compactness Principle and a comparison argument allow to verify that the problem possesses a nontrivial solution.
171 citations