Results on the Existence and Multiplicity of Solutions for a Class of Sublinear Degenerate Schrödinger Equations in $\mathbb{R}^N$

In this paper, we study the existence and multiplicity of nontrivial solutions of the semilinear degenerate Schrödinger equation
$$ -\mathcal{L}u + V(x)u = f(x,u),\qquad x\in \mathbb{R}^N,\quad N\ge 3, $$
where $V$ is a potential function defined on $\mathbb{R}^N$ and the nonlinearity $f$ is of sublinear growth and satisfies some appropriate conditions to be specified later. Here $\mathcal{L}$ is an $X$-elliptic operator with respect to a family $X = \{X_1, \ldots, X_m\}$ of locally Lipschitz continuous vector fields. We apply the Ekeland variational principle and a version of the fountain theorem in the proofs of our main existence results. Our main results extend and improve some recent ones in the literature.