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).
Mediterranean Journal of Mathematics – Springer Journals
Published: May 31, 2018
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