This paper, along with a subsequent one, investigates the existence of nontrivial solutions for some semi-linear elliptic equations in $ \mathbb{R}^N $. These problems are motivated by the search for certain types of solitary waves in nonlinear equations of the Klein-Gordon or Schrödinger type. The main equation considered is the nonlinear Klein-Gordon equation, which leads to the study of the equation $ -\Delta u + m u = f(u) $ in $ \mathbb{R}^N $, where $ m = a^2 - \omega^2 $. The function $ f $ is assumed to be real, continuous, odd, and $ f(0) = 0 $. The Lagrangian associated with this equation is expressed in terms of $ u $, and the problem is to find nontrivial solutions $ u \in H^1(\mathbb{R}^N) $ such that the Lagrangian is finite. This requires $ u $ to vanish at infinity, which is imposed as a boundary condition. The paper also considers traveling wave solutions, leading to a different elliptic equation. Similarly, stationary states of nonlinear Schrödinger equations lead to the same problem. The main result is the existence of nontrivial solutions for the semi-linear elliptic problem $ -\Delta u = g(u) $ in $ \mathbb{R}^N $, where $ g $ is a continuous, odd function with $ g(0) = 0 $. The paper shows that under these conditions, nontrivial solutions exist, and the solutions have exponential decay at infinity. The study is part of a broader investigation into the existence of solutions for semi-linear elliptic equations.This paper, along with a subsequent one, investigates the existence of nontrivial solutions for some semi-linear elliptic equations in $ \mathbb{R}^N $. These problems are motivated by the search for certain types of solitary waves in nonlinear equations of the Klein-Gordon or Schrödinger type. The main equation considered is the nonlinear Klein-Gordon equation, which leads to the study of the equation $ -\Delta u + m u = f(u) $ in $ \mathbb{R}^N $, where $ m = a^2 - \omega^2 $. The function $ f $ is assumed to be real, continuous, odd, and $ f(0) = 0 $. The Lagrangian associated with this equation is expressed in terms of $ u $, and the problem is to find nontrivial solutions $ u \in H^1(\mathbb{R}^N) $ such that the Lagrangian is finite. This requires $ u $ to vanish at infinity, which is imposed as a boundary condition. The paper also considers traveling wave solutions, leading to a different elliptic equation. Similarly, stationary states of nonlinear Schrödinger equations lead to the same problem. The main result is the existence of nontrivial solutions for the semi-linear elliptic problem $ -\Delta u = g(u) $ in $ \mathbb{R}^N $, where $ g $ is a continuous, odd function with $ g(0) = 0 $. The paper shows that under these conditions, nontrivial solutions exist, and the solutions have exponential decay at infinity. The study is part of a broader investigation into the existence of solutions for semi-linear elliptic equations.