Infinitely Many Solutions of Nonlocal Kirchhoff-Type Equations via Perturbation Methods

We study the multiplicity of weak solutions to the boundary-value problem
\begin{alignat}{2} - M\biggl(\iint_{\mathbb R^{2N}}|u(x)-u(y)|^2 K(x-y)\,d x\,d y\biggr)\mathscr L^s_K u &= f(x,u)+ g(x,u)&\qquad &\text{in}\quad \Omega,\nonumber \\ u&=0 &\qquad &\text{in}\quad \mathbb R^N\backslash \Omega, \nonumber \end{alignat}
where $\mathscr L^s_K$ is a nonlocal operator with singular kernel $K$, $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$ with dimension $N>2s$, parameter $s\in (0,1)$, $M$ is continuous function and $f(\cdot,\xi)$ is odd in $\xi$, $g(\cdot,\xi)$ is a perturbation term. By using the perturbation method of Rabinowitz, we show that there are infinitely many weak solutions to the problem.