Browsing by Author "Miyagaki, Olimpio Hiroshi"
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Item Asymptotic behavior of ground states of generalized pseudo-relativistic Hartree equation.(2019) Belchior, Pedro; Bueno, Hamilton Prado; Miyagaki, Olimpio Hiroshi; Pereira, Gilberto de AssisAbstract. With appropriate hypotheses on the nonlinearity f , we prove the existence of a ground state solution u for the problem − + m2u + V u = W ∗ F (u) f (u) in RN, where V is a bounded potential, not necessarily continuous, and F the primitive of f . We also show that any of this problem is a classical solution. Furthermore, we prove that the ground state solution has exponential decay.Item Existence and multiplicity of solutions for a supercritical elliptic problem in unbounded cylinders.(2017) Assunção, Ronaldo Brasileiro; Miyagaki, Olimpio Hiroshi; Rodrigues, Bruno MendesWe consider the following elliptic problem: – div( |∇u| p–2∇u |y| ap ) = |u| q–2u |y| bq + f(x) in , u = 0 on ∂, in an unbounded cylindrical domain := (y,z) ∈ Rm+1 × RN–m–1;0< A < |y| < B < ∞ , where 1 ≤ m < N – p, q = q(a, b) := Np N–p(a+1–b) , p > 1 and A, B ∈ R+. Let p∗ N,m := p(N–m) N–m–p . We show that p∗ N,m is the true critical exponent for this problem. The starting point for a variational approach to this problem is the known Maz’ja’s inequality (Sobolev Spaces, 1980) which guarantees, for the q previously defined, that the energy functional associated with this problem is well defined. This inequality generalizes the inequalities of Sobolev (p = 2, a = 0 and b = 0) and Hardy (p = 2, a = 0 and b = 1). Under certain conditions on the parameters a and b, using the principle of symmetric criticality and variational methods, we prove that the problem has at least one solution in the case f ≡ 0 and at least two solutions in the case f ≡ 0, if p < q < p∗ N,m.Item Existence and multiplicity results for an elliptic problem involving cylindrical weights and a homogeneous term μ.(2019) Assunção, Ronaldo Brasileiro; Miyagaki, Olimpio Hiroshi; Leme, Leandro Correia Paes; Rodrigues, Bruno MendesWe consider the following elliptic problem ⎧⎨ ⎩ − div |∇u| p−2 ∇u |y| ap = μ |u| p−2 u |y| p(a+1) + h(x) |u| q−2 u |y| bq + f(x, u) in Ω, u = 0 on ∂Ω, in an unbounded cylindrical domain Ω := {(y, z) ∈ Rm+1 × RN−m−1 ; 0 1, 1 ≤ mItem Multiplicity of positive solutions for the Kirchhoff-type equations with critical exponent in RN.(2018) Miyagaki, Olimpio Hiroshi; Leme, Leandro Correia Paes; Rodrigues, Bruno MendesIn this work we study the existence and multiplicity of solutions to the following Kirchhofftype problem with critical nonlinearity in RN ⎧⎨ ⎩ − ( a + b ∫ RN |∇u|pdx ) Δpu = μup∗−1 + λf (x, u); x ∈ RN , u ∈ D1,p(RN ), where N ≥ 2p, μ, λ, a, b > 0 and the nonlinearity f (x, u) satisfies certain subcritical growth conditions. By using topological and variational methods, infinitely many positive solutions are obtained.Item On a class of nonhomogeneous equations of Hénon-type : symmetry breaking and non radial solutions.(2017) Assunção, Ronaldo Brasileiro; Miyagaki, Olimpio Hiroshi; Pereira, Gilberto de Assis; Rodrigues, Bruno MendesIn this work we study the following Hénon-type equation ⎧⎪⎨ ⎪⎩ −div ( |∇u|p−2∇u |x|ap ) = |x|βf(u), in B; u > 0, in B; u = 0, on ∂B; where B := { x ∈ RN ; |x| < 1 } is a ball centered at the origin, the parameters verify the inequalities 0 ≤ a < N−p p , N ≥ 4, β > 0, 2 ≤ p < Np+pβ N−p(a+1) , and the nonlinearity f is nonhomogeneous. By minimization on the Nehari manifold, we prove that for large values of the parameter β there is a symmetry breaking and non radial solutions appear.Item Remarks about a generalized pseudo-relativistic Hartree equation.(2019) Bueno, Hamilton Prado; Miyagaki, Olimpio Hiroshi; Pereira, Gilberto de AssisWith appropriate hypotheses on the nonlinearity f , we prove the existence of a ground state solution u for the problem (− + m2) σ u + V u = (W ∗ F (u)) f (u) in RN, where 0 <σ < 1, V is a bounded continuous potential and F the primitive of f . We also show results about the regularity of any solution of this problem.Item Singular nonhomogeneous quasilinear elliptic equations with a convection term.(2017) Gonçalves, Jose Valdo Abreu; Marcial, Marcos Roberto; Miyagaki, Olimpio HiroshiIn this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub-super solution method, fixed point theorem, Leray–Schauder degree theory and comparison theorems.Item Topological structure of the solution set of singular equations with sign changing terms under dirichlet boundary condition.(2015) Gonçalves, José Vicente; Marcial, Marcos Roberto; Miyagaki, Olimpio HiroshiIn this paper we establish existence of connected components of positive solutions of the equation −∆pu = λf(u) in Ω, under Dirichlet boundary conditions, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, ∆p is the p-Laplacian, and f : (0, ∞) → R is a continuous function which may blow up to ±∞ at the origin.