Multiple Solutions for Semilinear $\Delta_{\gamma}-$differential Equations in $\mathbb R^N$ with Sign-changing Potential

Abstract

In this paper, we study the existence of infinitely many nontrivial solutions of the semilinear $\Delta_{\gamma}$ differential equations in $\mathbb{R}^N$ $$ - \Delta_{\gamma}u+ b(x)u=f(x,u)\quad \mbox{ in }\; \mathbb{R}^N, \quad u \in S^2_{\gamma}(\mathbb{R}^N), $$ where $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}: =\frac{\partial }{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, ..., \gamma_N),$$ and the potential $b$ is allowed to be sign-changing, and the primitive of the nonlinearity $f$ is of superquadratic growth near infinity in $u$ and allowed to be sign-changing.