Positive solutions for semilinear fractional elliptic problems involving an inverse fractional operator. (February 2020) - Álvarez-Caudevilla, P.; Colorado, E.; Ortega, Alejandro
Abstract: This paper is devoted to the study of the existence of positive solutions for a problem related to a higher order fractional differential equation involving a nonlinear term depending on a fractional differential operator, ( − Δ ) α u = λ u + ( − Δ ) β | u | p − 1 u i n Ω, ( − Δ ) j u = 0 o n ∂ Ω, f o r j ∈ Z, 0 ≤ j < [ α ], where Ω is a bounded domain in R N, 0 < β < 1, β < α < β + 1 and λ > 0 . In particular, we study the fractional elliptic problem, ( − Δ ) α − β u = λ ( − Δ ) − β u + | u | p − 1 u i n Ω, u = 0 o n ∂ Ω, and we prove existence or nonexistence of positive solutions depending on the parameter λ > 0, up to the critical value of the exponent p, i.e., for 1 < p ≤ 2 μ ∗ − 1 where μ ≔ α − β and 2 μ ∗ = 2 N N − 2 μ is the critical exponent of the Sobolev embedding.