Local behavior and existence of solutions for problems involving fractional (p,q)- Laplacian
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Universidade Federal de Minas Gerais
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Tese de doutorado
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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.
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Laplacian, Differential equation