This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator (- Δ) s and involving a critical Sobolev term. In particular, we consider(-Δ)su=γ|u|2∗-2u+f(x,u)inΩu=0inRnΩ,where Ω ⊂ Rn is an open bounded set with continuous boundary, n> 2 s with s∈ (0 , 1) , γ is a positive real parameter, 2 ∗= 2 n/ (n- 2 s) is the fractional critical Sobolev exponent and f is a Carathéodory function satisfying different subcritical conditions. For this problem we prove two different results of multiple solutions in the case when f is an odd function. When f has not any symmetry it is still possible to get a multiplicity result: we show that the problem under consideration admits at least two solutions of different sign.
Fiscella, A., Molica Bisci, G., Servadei, R. (2018). Multiplicity results for fractional Laplace problems with critical growth. MANUSCRIPTA MATHEMATICA, 155(3-4), 369-388 [10.1007/s00229-017-0947-2].
Multiplicity results for fractional Laplace problems with critical growth
Fiscella A;
2018
Abstract
This paper deals with multiplicity and bifurcation results for nonlinear problems driven by the fractional Laplace operator (- Δ) s and involving a critical Sobolev term. In particular, we consider(-Δ)su=γ|u|2∗-2u+f(x,u)inΩu=0inRnΩ,where Ω ⊂ Rn is an open bounded set with continuous boundary, n> 2 s with s∈ (0 , 1) , γ is a positive real parameter, 2 ∗= 2 n/ (n- 2 s) is the fractional critical Sobolev exponent and f is a Carathéodory function satisfying different subcritical conditions. For this problem we prove two different results of multiple solutions in the case when f is an odd function. When f has not any symmetry it is still possible to get a multiplicity result: we show that the problem under consideration admits at least two solutions of different sign.
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