Periodic solutions for critical fractional problems

Periodic solutions for critical fractional problems We deal with the existence of $$2\pi $$ 2 π -periodic solutions to the following non-local critical problem $$\begin{aligned} \left\{ \begin{array}{ll} [(-\Delta _{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &{}\quad \text{ in } \; (-\pi ,\pi )^{N} \\ u(x+2\pi e_{i})=u(x) &{}\quad \text{ for } \text{ all } \; x \in \mathbb {R}^{N}, \quad i=1, \dots , N, \end{array} \right. \end{aligned}$$ [ ( - Δ x + m 2 ) s - m 2 s ] u = W ( x ) | u | 2 s ∗ - 2 u + f ( x , u ) in ( - π , π ) N u ( x + 2 π e i ) = u ( x ) for all x ∈ R N , i = 1 , ⋯ , N , where $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$N \ge 4s$$ N ≥ 4 s , $$m\ge 0$$ m ≥ 0 , $$2^{*}_{s}=\frac{2N}{N-2s}$$ 2 s ∗ = 2 N N - 2 s is the fractional critical Sobolev exponent, W(x) is a positive continuous function, and f(x, u) is a superlinear $$2\pi $$ 2 π -periodic (in x) continuous function with subcritical growth. When $$m>0$$ m > 0 , the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder $$(-\,\pi ,\pi )^{N}\times (0, \infty )$$ ( - π , π ) N × ( 0 , ∞ ) , with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case $$m=0$$ m = 0 by using a careful procedure of limit. As far as we know, all these results are new. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

Periodic solutions for critical fractional problems

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-018-1317-y
Publisher site
See Article on Publisher Site

Abstract

We deal with the existence of $$2\pi $$ 2 π -periodic solutions to the following non-local critical problem $$\begin{aligned} \left\{ \begin{array}{ll} [(-\Delta _{x}+m^{2})^{s}-m^{2s}]u=W(x)|u|^{2^{*}_{s}-2}u+ f(x, u) &{}\quad \text{ in } \; (-\pi ,\pi )^{N} \\ u(x+2\pi e_{i})=u(x) &{}\quad \text{ for } \text{ all } \; x \in \mathbb {R}^{N}, \quad i=1, \dots , N, \end{array} \right. \end{aligned}$$ [ ( - Δ x + m 2 ) s - m 2 s ] u = W ( x ) | u | 2 s ∗ - 2 u + f ( x , u ) in ( - π , π ) N u ( x + 2 π e i ) = u ( x ) for all x ∈ R N , i = 1 , ⋯ , N , where $$s\in (0,1)$$ s ∈ ( 0 , 1 ) , $$N \ge 4s$$ N ≥ 4 s , $$m\ge 0$$ m ≥ 0 , $$2^{*}_{s}=\frac{2N}{N-2s}$$ 2 s ∗ = 2 N N - 2 s is the fractional critical Sobolev exponent, W(x) is a positive continuous function, and f(x, u) is a superlinear $$2\pi $$ 2 π -periodic (in x) continuous function with subcritical growth. When $$m>0$$ m > 0 , the existence of a nonconstant periodic solution is obtained by applying the Linking Theorem, after transforming the above non-local problem into a degenerate elliptic problem in the half-cylinder $$(-\,\pi ,\pi )^{N}\times (0, \infty )$$ ( - π , π ) N × ( 0 , ∞ ) , with a nonlinear Neumann boundary condition, through a suitable variant of the extension method in periodic setting. We also consider the case $$m=0$$ m = 0 by using a careful procedure of limit. As far as we know, all these results are new.

Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Feb 22, 2018

References

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