DSpace Coleção:
http://www.repositorio.ufop.br/jspui/handle/123456789/590
Wed, 15 Feb 2023 13:45:12 GMT2023-02-15T13:45:12ZOn a singular minimizing problem.
http://www.repositorio.ufop.br/jspui/handle/123456789/16136
Título: On a singular minimizing problem.
Autor(es): Ercole, Grey; Pereira, Gilberto de Assis
Resumo: For 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.Mon, 01 Jan 2018 00:00:00 GMThttp://www.repositorio.ufop.br/jspui/handle/123456789/161362018-01-01T00:00:00ZExtremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness.
http://www.repositorio.ufop.br/jspui/handle/123456789/16135
Título: Extremal solutions of strongly coupled nonlinear elliptic systems and L∞-boundedness.
Autor(es): Costa, Felipe; Souza, Gil Fidelix de; Montenegro, Marcos
Resumo: The paper concerns positive solutions for the Dirichlet problem −Lu = ΛF(x, u) in Ω, u = 0 on ∂Ω,
where Ω is a smooth bounded domain in Rn, n ≥ 2, u = (u1, ..., um) : Ω → Rm, m ≥ 1, Lu = (L1u1, ..., Lmum), where each Li denotes a uniformly elliptic linear operator of second order in nondivergence form in Ω, Λ = (λ1, ..., λm) ∈ Rm, F = (f1, ..., fm) : Ω × Rm → Rm and ΛF(x, u) = (λ1f1(x, u), ..., λmfm(x, u)).
For a general class of maps F we prove that there exists a hypersurface Λ∗ in Rm + := (0, ∞)m such that tuples Λ ∈ Rm + below Λ∗ correspond to minimal positive strong solutions of the above system. Stability of these solutions is also discussed. Already for tuples above Λ∗, there is no nonnegative strong solution. The shape of the hypersurface Λ∗ depends on growth on u of the nonlinearity F in a sense to be specified. When Λ ∈ Λ∗ and the coefficients of each operator Li are slightly smooth, the problem admits a unique minimal nonnegative weak solution, called extremal solution. Furthermore, when F depends only on u and all Li are Laplace operators, we investigate the L∞ regularity of this solution for any m ≥ 1 in dimensions 2 ≤ n ≤ 9 for balls and n = 2 and n = 3 for convex domains.Sat, 01 Jan 2022 00:00:00 GMThttp://www.repositorio.ufop.br/jspui/handle/123456789/161352022-01-01T00:00:00ZExistence of a positive solution for a class of non-local elliptic problem with critical growth in Rn.
http://www.repositorio.ufop.br/jspui/handle/123456789/16134
Título: Existence of a positive solution for a class of non-local elliptic problem with critical growth in Rn.
Autor(es): Leme, Leandro Correia Paes; Rodrigues, Bruno Mendes
Resumo: In this article, we consider the following non-local elliptic equation with critical growth ⎧⎪⎨⎪⎩−
a + b RN |∇u| 2 dx p−1 2 Δu = λk(x)uq + u2∗−1, x ∈ RN , u ∈ D1,2(RN ), where N ≥ 3, λ > 0, 2∗:= 2N
N−2 , 1 < p ≤ q < 2∗ − 1, a ≥ 0, b ≥ 0 and k(x) ∈ L 2∗ 2∗−q−1 (RN ) is a nonnegative function. Using variational methods and concentration-compactness principle, we obtain a positive solution.Sat, 01 Jan 2022 00:00:00 GMThttp://www.repositorio.ufop.br/jspui/handle/123456789/161342022-01-01T00:00:00ZExistence and multiplicity results for an elliptic problem involving cylindrical weights and a homogeneous term μ.
http://www.repositorio.ufop.br/jspui/handle/123456789/16133
Título: Existence and multiplicity results for an elliptic problem involving cylindrical weights and a homogeneous term μ.
Autor(es): Assunção, Ronaldo Brasileiro; Miyagaki, Olimpio Hiroshi; Leme, Leandro Correia Paes; Rodrigues, Bruno Mendes
Resumo: We 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 <A< |y| <B< ∞}, where A, B ∈ R+, p > 1, 1 ≤ m<N − p, q := N p N − p(a + 1 − b), 0 ≤ μ < μ := m + 1 − p(a + 1) p p , h ∈ L N q (Ω) ∩ L∞(Ω) is a positive function and f : Ω × R → R is a Carath ́eodory function with growth at infinity. Using the Krasnoselski’s genus and applying Z2 version of the
Mountain Pass Theorem, we prove, under certain assumptions about f, that the above problem has infinite invariant solutions.Tue, 01 Jan 2019 00:00:00 GMThttp://www.repositorio.ufop.br/jspui/handle/123456789/161332019-01-01T00:00:00Z