Existence and Multiplicity of Solutions for Fractional Elliptic Problems with Discontinuous Nonlinearities

Existence and Multiplicity of Solutions for Fractional Elliptic Problems with Discontinuous... We consider the following fractional elliptic problem: $$\begin{aligned} (P)\left\{ \begin{array}{ll} (-\Delta )^s u = f(u) H(u-\mu )&{} \quad \text{ in } \ \Omega ,\\ u =0 &{}\quad \text{ on } \ \mathbb{{R}}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$ ( P ) ( - Δ ) s u = f ( u ) H ( u - μ ) in Ω , u = 0 on R n \ Ω , where $$(-\Delta )^s, s\in (0,1)$$ ( - Δ ) s , s ∈ ( 0 , 1 ) is the fractional Laplacian, $$\Omega $$ Ω is a bounded domain of $$\mathbb{{R}}^n,(n\ge 2s)$$ R n , ( n ≥ 2 s ) with smooth boundary $$\partial \Omega ,$$ ∂ Ω , H is the Heaviside step function, f is a given function and $$\mu $$ μ is a positive real parameter. The problem (P) can be considered as simplified version of some models arising in different contexts. We employ variational techniques to study the existence and multiplicity of positive solutions of problem (P). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Existence and Multiplicity of Solutions for Fractional Elliptic Problems with Discontinuous Nonlinearities

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-018-1188-7
Publisher site
See Article on Publisher Site

Abstract

We consider the following fractional elliptic problem: $$\begin{aligned} (P)\left\{ \begin{array}{ll} (-\Delta )^s u = f(u) H(u-\mu )&{} \quad \text{ in } \ \Omega ,\\ u =0 &{}\quad \text{ on } \ \mathbb{{R}}^n {\setminus } \Omega , \end{array} \right. \end{aligned}$$ ( P ) ( - Δ ) s u = f ( u ) H ( u - μ ) in Ω , u = 0 on R n \ Ω , where $$(-\Delta )^s, s\in (0,1)$$ ( - Δ ) s , s ∈ ( 0 , 1 ) is the fractional Laplacian, $$\Omega $$ Ω is a bounded domain of $$\mathbb{{R}}^n,(n\ge 2s)$$ R n , ( n ≥ 2 s ) with smooth boundary $$\partial \Omega ,$$ ∂ Ω , H is the Heaviside step function, f is a given function and $$\mu $$ μ is a positive real parameter. The problem (P) can be considered as simplified version of some models arising in different contexts. We employ variational techniques to study the existence and multiplicity of positive solutions of problem (P).

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: May 31, 2018

References

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