Multiplicity of solutions for a p-fractional Schrödinger-Kirchhoff type equation

We consider the problem
$$ -\left(a+b\left([u]_s^p\right)^{p-1}\right)(-\Delta)_p^s u(x)+V(x)|u(x)|^{p-2} u(x)=f(x, u(x)), $$
with \(|u(x)| \rightarrow 0\),as \(|x| \rightarrow+\infty,\) where \(s \in(0,1), 1<p<+\infty, a>0, b>0, N \geq 2,\) \(1<s p<N<+\infty, V \in C\left(\mathbb{R}^N\right)\) verifies \(\theta=\inf _{\mathbb{R}^N} V>0\), and \(f \in C\left(\mathbb{R}^N \times \mathbb{R}\right).\) The operator \((-\Delta)_p^s\) denotes the fractional \(p\)-Laplacian, and the seminorm \([u]_{s, p}^p\) is associated with the fractional Sobolev space \(W^{s, p}\left(\mathbb{R}^N\right)\). Our approach combines variational methods and critical point theory to establish the existence of nontrivial solutions. Under appropriate conditions on \(V\) and \(f\), we prove the existence of a ground-state solution. Moreover, by assuming an additional oddness condition on the nonlinearity, we obtain the multiplicity using the genus theory.