TY - JOUR

T1 - Fractional Laplacian equations with critical Sobolev exponent

AU - Servadei, Raffaella

AU - Valdinoci, Enrico

PY - 2015/9/12

Y1 - 2015/9/12

N2 - In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator LK (Formula Presented), that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here s∈(0,1),Ω is an open bounded set of Rn,n>2s, with continuous boundary, λ is a positive real parameter, 2∗=2n/(n-2s) is a fractional critical Sobolev exponent and f is a lower order perturbation of the critical power |u|2∗-2u, while LK is the integrodifferential operator defined as (Formula Presented). Under suitable growth condition on f, we show that this problem admits non-trivial solutions for any positive parameter λ. This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when K(x)=|x|-(n+2s) (this gives rise to the fractional Laplace operator -(-Δ)s), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

AB - In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator LK (Formula Presented), that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here s∈(0,1),Ω is an open bounded set of Rn,n>2s, with continuous boundary, λ is a positive real parameter, 2∗=2n/(n-2s) is a fractional critical Sobolev exponent and f is a lower order perturbation of the critical power |u|2∗-2u, while LK is the integrodifferential operator defined as (Formula Presented). Under suitable growth condition on f, we show that this problem admits non-trivial solutions for any positive parameter λ. This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when K(x)=|x|-(n+2s) (this gives rise to the fractional Laplace operator -(-Δ)s), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

KW - Best fractional critical Sobolev constant

KW - Critical nonlinearities

KW - Fractional Laplacian

KW - Integrodifferential operators

KW - Linking Theorem

KW - Mountain Pass Theorem

KW - Palais–Smale condition

KW - Variational techniques

UR - http://www.scopus.com/inward/record.url?scp=84938957047&partnerID=8YFLogxK

U2 - 10.1007/s13163-015-0170-1

DO - 10.1007/s13163-015-0170-1

M3 - Article

AN - SCOPUS:84938957047

VL - 28

SP - 655

EP - 676

JO - Revista Matematica Complutense

JF - Revista Matematica Complutense

SN - 1139-1138

IS - 3

ER -