This paper, dedicated to Klaus Kirchgässner on his 60th birthday, is motivated by recent studies on nonlinear Schrödinger equations. The authors build upon the work of Floer and Weinstein [1], who used a Lyapunov-Schmidt reduction to prove the existence of standing wave solutions for a specific form of the nonlinear Schrödinger equation. Oh [2-3] extended this work to a more general equation, allowing for a broader class of potentials. However, these studies require that the potential \( V \) belongs to a specific class \( (V)_a \), which excludes rapidly oscillating potentials. The goal of this paper is to use variational methods based on variants of the Mountain Pass Theorem to derive existence results for a more general class of potentials, including highly oscillatory ones that do not belong to \( (V)_a \). The authors aim to show that their methods can handle potentials with arbitrary oscillation patterns, though they cannot guarantee the concentration of the solution's support. The paper introduces a more general semilinear elliptic PDE and outlines the hypotheses for the functions involved, ensuring the existence of solutions under certain conditions.This paper, dedicated to Klaus Kirchgässner on his 60th birthday, is motivated by recent studies on nonlinear Schrödinger equations. The authors build upon the work of Floer and Weinstein [1], who used a Lyapunov-Schmidt reduction to prove the existence of standing wave solutions for a specific form of the nonlinear Schrödinger equation. Oh [2-3] extended this work to a more general equation, allowing for a broader class of potentials. However, these studies require that the potential \( V \) belongs to a specific class \( (V)_a \), which excludes rapidly oscillating potentials. The goal of this paper is to use variational methods based on variants of the Mountain Pass Theorem to derive existence results for a more general class of potentials, including highly oscillatory ones that do not belong to \( (V)_a \). The authors aim to show that their methods can handle potentials with arbitrary oscillation patterns, though they cannot guarantee the concentration of the solution's support. The paper introduces a more general semilinear elliptic PDE and outlines the hypotheses for the functions involved, ensuring the existence of solutions under certain conditions.