Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction (Q2082152)

The authors study the following fractional Kirchhoff equation with discontinuous nonlinearity \[ \begin{cases} \left(\varepsilon^{2\alpha}a+\varepsilon^{4\alpha-3}b\int\limits_{\mathbb{R}^3}\left|\left(-\Delta\right)^{\frac{\alpha}{2}}u\right|^2 \,\mathrm{d}x\right) \left(-\Delta\right)^\alpha u +V(x)u=H(u-\beta)f(u)\quad\text{in }\mathbb{R}^3,\\ u\in H^\alpha(\mathbb{R}^3), \quad u>0, \end{cases} \] where \(\varepsilon, \beta>0\) are small parameters, \(\alpha \in (\frac{3}{4},1)\), \(a,b\) are positive constants and the function \(f\) is superlinear with subcritical growth. Based on minimax theorems, existence and concentration properties of positive solutions are shown to the problem above.