Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction

Abstract

AbstractIn this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} \left( \varepsilon ^{2\alpha }a+\varepsilon ^{4\alpha -3}b\int _{{\mathbb {R}}^3}|(-\Delta )^{\frac{\alpha }{2}} u|^2{{\mathrm{d}}}x\right) (-\Delta )^\alpha {u}+V(x)u = H(u-\beta )f(u) &{} \quad \text{ in }\,\,{\mathbb {R}}^3, \\ u\in H^\alpha ({\mathbb {R}}^3),\quad u>0 &{} \quad \text{ in }\,\, {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$ ε 2 α a + ε 4 α - 3 b ∫ R 3 | ( - Δ ) α 2 u | 2 d x ( - Δ ) α u + V ( x ) u = H ( u - β ) f ( u ) in R 3 , u ∈ H α ( R 3 ) , u > 0 in R 3 , where $$\varepsilon ,\beta >0$$ ε , β > 0 are small parameters, $$\alpha \in (\frac{3}{4},1)$$ α ∈ ( 3 4 , 1 ) and a, b are positive constants, $$(-\Delta )^{\alpha }$$ ( - Δ ) α is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system.