Fučik spectrum of fractional Schrödinger operator with the unbounded potential on \(\mathbb {R}^{N}\)

In this paper, we are concerned with the Fučik spectrum of fractional Schrödinger operators \({\varvec{(-\Delta )^{s}+V}}\) ${\mathbf{(}\mathbf{-}\mathbf{\Delta }\mathbf{)}}^{\mathit{s}}\mathbf{+}\mathit{V}$ , which is defined as the set \({\varvec{\Sigma _{K}}}\) ${\mathbf{\Sigma }}_{\mathit{K}}$ of all \({\varvec{(\alpha , \beta ) \in \mathbb {R}^{2}}}\) $\mathbf{(}\mathit{\alpha }\mathbf{,}\mathit{\beta }\mathbf{)}\mathbf{\in }{\mathbb{R}}^{\mathbf{2}}$ such that \(\begin{aligned} {\varvec{(-\Delta )^{s} u+V(x) u=\alpha u^{+}-\beta u^{-}, \quad x \in \mathbb {R}^{N}}} \end{aligned}\) $$\begin{array}{c}\hfill {\mathbf{(}\mathbf{-}\mathbf{\Delta }\mathbf{)}}^{\mathit{s}}\mathit{u}\mathbf{+}\mathit{V}\mathbf{(}\mathit{x}\mathbf{)}\mathit{u}\mathbf{=}\mathit{\alpha }{\mathit{u}}^{\mathbf{+}}\mathbf{-}\mathit{\beta }{\mathit{u}}^{\mathbf{-}}\mathbf{,}\mathit{x}\mathbf{\in }{\mathbb{R}}^{\mathit{N}}\end{array}$$ has a non-trivial solution \({\varvec{u}}\) $\mathit{u}$ , where \({\varvec{u^{\pm }=\max \{\pm u, 0\}}}\) ${\mathit{u}}^{\mathbf{\pm }}\mathbf{=}\mathbf{max}\mathbf{\{}\mathbf{\pm }\mathit{u}\mathbf{,}\mathbf{0}\mathbf{\}}$ , \({\varvec{N>2s}}\) $\mathit{N}\mathbf{>}\mathbf{2}\mathit{s}$ , \({\varvec{s \in (0,1)}}\) $\mathit{s}\mathbf{\in }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{)}$ and V is an external unbounded potential function. Based on variational method, it turns out that the Fučik spectrum has a first non-trivial curve \({\varvec{\mathcal {C}}}\) $\mathcal{C}$ being Lipschitz continuous, decreasing and with a certain asymptotic behavior. And then, as applications, we obtain the existence of non-trivial solutions for nonlinear Schrödinger equations with non-resonant nonlinearity.