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
SN - 1139-1138
VL - 28
SP - 655
EP - 676
JO - Revista Matematica Complutense
JF - Revista Matematica Complutense
IS - 3
ER -