This paper, authored by H. Berestycki and P.-L. Lions, focuses on the existence of nontrivial solutions for semi-linear elliptic equations in \(\mathbb{R}^N\). These equations are motivated by the search for solitary waves (stationary states) in nonlinear Klein-Gordon and Schrödinger equations. The main equation considered is: \[ \Phi_{tt} - \Delta \Phi + a^2 \Phi = f(\Phi), \] where \(\Phi = \Phi(t, x)\) is a complex function, \(\Delta \Phi\) is the Laplacian, and \(a\) is a real constant. The function \(f\) is assumed to be odd and satisfy \(f(0) = 0\). The Lagrangian density for this equation is: \[ \mathcal{L}_\Phi = -\frac{1}{2} |\Phi_t|^2 + \frac{1}{2} |\nabla \Phi|^2 + \frac{a^2}{2} |\Phi|^2 - F(|\Phi|), \] where \(F(\rho) = \int_0^\rho f(s) \, ds\). For solitary wave solutions of the form \(\Phi(t, x) = e^{i\omega t} u(x)\), the equation reduces to: \[ -\Delta u + mu = f(u) \text{ in } \mathbb{R}^N, \] where \(m = a^2 - \omega^2\). The Lagrangian \(S(u)\) is given by: \[ S(u) = \int_{\mathbb{R}^N} \mathcal{L}_\Phi \, dx = \frac{1}{2} \int_{\mathbb{R}^N} |\nabla u|^2 \, dx + \frac{m}{2} \int_{\mathbb{R}^N} u^2 \, dx - \int_{\mathbb{R}^N} F(u) \, dx. \] The authors impose the condition \(u \in H^1(\mathbb{R}^N)\) to ensure that the Lagrangian is finite and \(u\) vanishes at infinity. They also consider traveling wave solutions and stationary states of nonlinear Schrödinger equations, which lead to similar semi-linear elliptic problems: \[ -\Delta u = g(u) \text{ in } \mathbb{R}^N, \] where \(g : \mathbb{R} \to \mathbb{R}\) is a continuous, odd function with \(g(0) = 0\).This paper, authored by H. Berestycki and P.-L. Lions, focuses on the existence of nontrivial solutions for semi-linear elliptic equations in \(\mathbb{R}^N\). These equations are motivated by the search for solitary waves (stationary states) in nonlinear Klein-Gordon and Schrödinger equations. The main equation considered is: \[ \Phi_{tt} - \Delta \Phi + a^2 \Phi = f(\Phi), \] where \(\Phi = \Phi(t, x)\) is a complex function, \(\Delta \Phi\) is the Laplacian, and \(a\) is a real constant. The function \(f\) is assumed to be odd and satisfy \(f(0) = 0\). The Lagrangian density for this equation is: \[ \mathcal{L}_\Phi = -\frac{1}{2} |\Phi_t|^2 + \frac{1}{2} |\nabla \Phi|^2 + \frac{a^2}{2} |\Phi|^2 - F(|\Phi|), \] where \(F(\rho) = \int_0^\rho f(s) \, ds\). For solitary wave solutions of the form \(\Phi(t, x) = e^{i\omega t} u(x)\), the equation reduces to: \[ -\Delta u + mu = f(u) \text{ in } \mathbb{R}^N, \] where \(m = a^2 - \omega^2\). The Lagrangian \(S(u)\) is given by: \[ S(u) = \int_{\mathbb{R}^N} \mathcal{L}_\Phi \, dx = \frac{1}{2} \int_{\mathbb{R}^N} |\nabla u|^2 \, dx + \frac{m}{2} \int_{\mathbb{R}^N} u^2 \, dx - \int_{\mathbb{R}^N} F(u) \, dx. \] The authors impose the condition \(u \in H^1(\mathbb{R}^N)\) to ensure that the Lagrangian is finite and \(u\) vanishes at infinity. They also consider traveling wave solutions and stationary states of nonlinear Schrödinger equations, which lead to similar semi-linear elliptic problems: \[ -\Delta u = g(u) \text{ in } \mathbb{R}^N, \] where \(g : \mathbb{R} \to \mathbb{R}\) is a continuous, odd function with \(g(0) = 0\).