Concentration of solutions for fractional Kirchhoff equations with discontinuous reaction
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.
scientific article; zbMATH DE number 7595887
scientific article