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Master's Dissertation
DOI
10.11606/D.55.2016.tde-10102016-163017
Document
Author
Full name
Maycon Sullivan Santos Araújo
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2015
Supervisor
Committee
Massa, Eugenio Tommaso (President)
Calanchi, Marta
Paiva, Francisco Odair Vieira de
Soares, Sérgio Henrique Monari
Title in Portuguese
Equações elípticas com não lineradidades críticas e perturbações de ordem inferior
Keywords in Portuguese
Equações diferenciais parciais
Equações diferenciais parciais elípticas
Problemas com expoente crítico
Abstract in Portuguese
Neste trabalho, tivemos como objetivo estudar a existência de soluções fracas não triviais para o problema elíptico com não linearidade crítica { - Δu = λu + u2* - 1+ + g(x, u+) + f(x); em Ω u = 0; sobre ∂ Ω , (P) onde Ω é um domínio limitado com fronteira suave em ℝN, com N ≥ 3, 2* = 2N / (N - 2) é o expoente crítico de Sobolev, u+ = max(u; 0), g ∈ C(Ω̄ x ℝ, ℝ+), λ > λ1, λ ∉ σ (- Δ) e f ∈ Lr> (Ω), com r > N. Com o intuito de observar as mudanças que ocorrem do caso subcrítico para o crítico e as diferentes técnicas variacionais para a resolução de problemas elípticos, estudamos, inicialmente, um problema um pouco mais antigo que (P), que, por sua vez, motivou seu estudo. Tal problema é { - Δu = λ u + up+ +f; em Ω u = 0; sobre ∂ Ω(P') onde consideramos o caso subcrítico, ou seja, quando p ∈ (1; 2* - 1). Com o auxílio do TEOREMA DE ENLACE verificamos que tanto (P) quanto (P') têm pelo menos duas soluções fracas não triviais.
Title in English
Eliptic equations with nonlinearities and critical order disturbances lower
Keywords in English
Elliptic partial differential equations
Partial differential equations
Problems with critical exponent
Abstract in English
In this work, we aimed to study the existence of nontrivial weak solutions for the elliptic problem with critical non-linearity { - Δu = λu + u2* - 1+ + g(x, u+) + f(x); in Ω u = 0; on ∂ Ω , (P) where Ω is a bounded domain with smooth boundary in ℝN, with N ≥ 3, 2* = 2N / N -2 is the critical Sobolev exponent, u+ = max(u; 0), g ∈ C(Ω̄ x ℝ, ℝ+), λ > λ1, λ ∉ σ (- Δ) and f ∈ Lr (Ω), with r > N. In order to observe different variational techniques for solving elliptic problems, we studied initially a problem a little older than (P), which, in turn, led to its study. This problem is { - Δu = λ u + up+ +f; inΩ u = 0; on ∂ Ω(P') where we consider the subcritical case, that is, when p ∈ (1, 2* - 1). With the aid of the LINKING THEOREM we see that both (P) and (P') have at least two nontrivial weak solutions.
 
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Publishing Date
2016-10-11
 
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