Local behavior and existence of solutions for problems involving fractional (p,q)- Laplacian

• Ano de defesa: 2019
• Autor(a) principal: Aldo Henrique de Souza Medeiros
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de Minas Gerais; Brasil; Programa de Pós-Graduação em Matemática; UFMG
• Palavras-chave em Português: Laplacian; Differential equation
• Link de acesso: http://hdl.handle.net/1843/30804

In the first part of this work, we study the regularity of weak solutions (in an appropriate space) of the elliptic partial differential equation (−∆p)su + (−∆q)su = f(x) in RN, where 0 < s < 1 and 2 ≤ q ≤ p < N/s, and we prove that these solutions are locally in C0,α(RN). In the sequence, we prove the existence of solutions of the problem (−∆p)su + (−∆q)su = |u|p∗ s−2u + λg(x)|u|r−2u in RN, where 1 < q ≤ p < N/s, λ is a parameter and g satisfies some integrability conditions. As an application of the previus result, we show that, if 0 < s < 1, 2 ≤ q ≤ p < N/s and g is bounded, then the obtained solutions are continuous and bounded. In the final part of the work, we study the behavior as p →∞ of up, a positive least energy solution of the problem        h(−∆p)α +−∆q(p) βiu = µpkukp−2 ∞ u(xu)δxu in Ω u = 0 in RN \Ω |u(xu)| = kuk∞, where Ω ⊂RN is a smooth bounded domain, δxu is the Dirac delta distribution supported at xu, lim p→∞ q(p) p = Q ∈((0,1) if 0 < β < α < 1 (1,∞) if 0 < α < β < 1 and lim p→∞ p √µp > R−α, with R denoting the inradius of Ω.