Abstract:
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.
