Browsing by Author "Pereira, Gilberto de Assis"
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Item An optimal pointwise Morrey-Sobolev inequality.(2020) Ercole, Grey; Pereira, Gilberto de AssisLet Ω be a bounded, smooth domain of RN , N ≥ 1. For each p > N we study the optimal function s = sp in the pointwise inequality |v(x)| ≤ s(x) ∇vLp(Ω) , ∀ (x, v) ∈ Ω × W1,p 0 (Ω). We show that sp ∈ C0,1−(N/p) 0 (Ω) and that sp converges pointwise to the distance function to the boundary, as p → ∞. Moreover, we prove that if Ω is convex, then sp is concave and has a unique maximum point.Item Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems.(2019) Ercole, Grey; Pereira, Gilberto de Assis; Sanchis, Remy de PaivaLet be a smooth, bounded domain of RN , ω be a positive, L1-normalized function, and 0 < s < 1 < p. We study the asymptotic behavior, as p → ∞, of the pair p p, u p, where p is the best constant C in the Sobolev-type inequality C exp (log |u| p)ωdx ≤ [u] p s,p ∀ u ∈ Ws,p 0 () and u p is the positive, suitably normalized extremal function corresponding to p. We show that the limit pairs are closely related to the problem of minimizing the quotient |u|s / exp (log |u|)ωdx , where |u|s denotes the s-Hölder seminorm of a function u ∈ C0,s 0 ().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 Asymptotic behaviour as p → ∞ of least energy solutions of a (p, q(p))-Laplacian problem.(2019) Alves, Claudianor Oliveira; Ercole, Grey; Pereira, Gilberto de AssisWe study the asymptotic behaviour, as p → ∞, of the least energy solutions of the problem −(Δp + Δq(p))u = λp|u(xu)| p−2u(xu)δxu in Ω u = 0 on ∂Ω, where xu is the (unique) maximum point of |u|, δxu is the Dirac delta distribution supported at xu, limp→∞ q(p) p = Q ∈ (0, 1) if N 0 is such that min ∇u∞ u∞ : 0 ≡ u ∈ W1,∞(Ω) ∩ C0(Ω) limp→∞(λp) 1/p < ∞.Item Existence and nonexistence of solutions to nonlocal elliptic problems.(2022) Bueno, Hamilton Prado; Pereira, Gilberto de Assis; Silva, Edcarlos Domingos da; Ruviaro, RicardoIt is established existence and nonexistence of solutions to nonlocal elliptic problems involving the generalized pseudo-relativistic Hartree equation. Our arguments are based on variational methods together with a fine analysis on the Pohozaev identity.Item Fractional Sobolev inequalities associated with singular problems.(2018) Ercole, Grey; Pereira, Gilberto de AssisIn this paper we study Sobolev-type inequalities associated with singular problems for the fractional p-Laplacian operator in a bounded domain of RN , N ≥ 2.Item Ground state of a magnetic nonlinear Choquard equation.(2019) Bueno, Hamilton Prado; Mamani, Guido Gutierrez; Pereira, Gilberto de AssisWe consider the stationary magnetic nonlinear Choquard equation −(∇ + iA(x))2u + V(x)u = (1|x|α ∗ (|u|) ) f(|u|) |u| u, where A : RN → RN is a vector potential, V is a scalar potential, f : R → R and F is the primitive of f . Under mild hypotheses, we prove the existence of a ground state solution for this problem. We also prove a simple multiplicity result by applying Ljusternik–Schnirelmann methods.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 On a singular minimizing problem.(2018) Ercole, Grey; Pereira, Gilberto de AssisFor each q ∈ (0, 1) let λq(Ω) := inf k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and Z Ω |v| q dx = 1, where p > 1 and Ω is a bounded and smooth domain of R N , N ≥ 2. We first show that 0 < μ(Ω) := lim q→0+λq(Ω)|Ω| p q < ∞, where |Ω| = R Ω dx. Then, we prove that μ(Ω) = min (k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and lim q→0+ 1 |Ω| Z Ω |v| q dx 1 q = 1) and that μ(Ω) is reached by a function u ∈ W1,p 0 (Ω), which is positive in Ω, belongs to C 0,α(Ω), for some α ∈ (0, 1), and satisfies − div(|∇u| p−2 ∇u) = μ(Ω)|Ω| −1 u −1 in Ω, and Z Ω log udx = 0. We also show that μ(Ω)−1 is the best constant C in the following log-Sobolev type inequality exp 1 |Ω| Z Ω log |v| p dx ≤ C k∇vk p Lp(Ω) , v ∈ W1,p 0 (Ω) and that this inequality becomes an equality if, and only if, v is a scalar multiple of u and C = μ(Ω)−1.Item Pohozaev-type identities for a pseudo-relativistic schrodinger operator and applications.(2019) Bueno, Hamilton Prado; Pereira, Gilberto de Assis; Medeiros, Aldo Henrique de SouzaIn this paper we prove a Pohozaev-type identity for both the prob- lem (−∆ + m2 ) su = f(u) in RN and its harmonic extension to R N+1 + when 0 < s < 1. So, our setting includes the pseudo-relativistic operator √ −∆ + m2 and the results showed here are original, to the best of our knowledge. The identity is first obtained in the extension setting and then “translated” into the original problem. In order to do that, we develop a specific Fourier trans- form theory for the fractionary operator (−∆ + m2 ) s , which lead us to define a weak solution u of the original problem if the identity (S) Z RN (−∆ + m2 ) s/2u(−∆ + m2 ) s/2 vdx = Z RN f(u)vdx is satisfied by all v ∈ Hs (RN ). The obtained Pohozaev-type identity is then applied to prove both a result of nonexistence of solution to the case f(u) = |u| p−2u if p ≥ 2 ∗ s and a result of existence of a ground state, if f is modeled by κu3/(1+u 2 ), for a constant κ. In this last case, we apply the Nehari-Pohozaev manifold introduced by D. Ruiz. Finally, we prove that positive solutions of (−∆ + m2 ) su = f(u) are radially symmetric and decreasing with respect to the origin, if f is modeled by functions like t α, α ∈ (1, 2 ∗ s − 1) or tln t.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 Results on a strongly coupled, asymptotically linear pseudo-relativistic Schrödinger system : ground state, radial symmetry and Hölder regularity.(2022) Bueno, Hamilton Prado; Mamani, Guido Gutierrez; Medeiros, Aldo Henrique de Souza; Pereira, Gilberto de AssisIn this paper we consider the asymptotically linear, strongly coupled nonlinear system ⎧ ⎪⎨ ⎪⎩ √ −∆ + m2 u = u 2 + v 2 1 + s(u2 + v 2) u + λv, √ −∆ + m2 v = u 2 + v 2 1 + s(u2 + v 2) v + λu, where m > 0, 0 < λ < m and 0 < s < 1/(λ + m) are constants. By applying the Nehari–Pohozaev manifold, we prove that our system has a ground state solution. We also prove that solutions of this system are radially symmetric and belong to C0,μ(RN ) for some 0 < μ < 1 and each N > 1.