A new linking theorem for Lipschitz functionals and its application

In this paper, we establish a new linking theorem for local Lipschitz functionals without the \(\tau \) $\tau$ -upper semi-continuity assumption. As an application, we study the following equation with strongly indefinite structure and sign-changing discontinuous nonlinearity \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + V(x)u=|u|^{p-2}u-\mu H_e(a-|u|)|u|^{q-2}u, \quad \text {a.e. in } \mathbb {R}^{N}, \\ u\in H^{1}(\mathbb {R}^{N}) , N\ge 2 , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+V(x)u={|u|}^{p-2}u-\mu H_e{(a-|u|)|u|}^{q-2}u,\text{a.e. in}{\mathbb{R}}^N,\hfill \\ u\in H^1({\mathbb{R}}^N),N\ge 2,\hfill \end{array}\right)\end{array}$$ where V is 1-periodic and 0 lies in a gap of the spectrum of \(-\Delta +V\) $-\mathrm{\Delta }+V$ , \(2<q<p<2^*\) $2<q<p<2^{\ast }$ and \(\mu , a >0\) $\mu ,a>0$ are parameters. We prove the existence of a nontrivial solution of the equation above when \(\mu >0\) $\mu >0$ is small enough.