Gradient Lagrangian systems and semilinear PDE

Abstract

We survey some results about multiplicity of certain classes of entire solutions to semilinear elliptic equations or systems of the form $-\Delta u = F_{u}(x, u)$, $x\in\mathbb{R}^{N+1}$, including the Allen Cahn or the stationary Nonlinear Schr\"odinger case. In connection with this kind of problems we study some metric separation properties of sublevels of the functional $V(u) = \tfrac 12\|\nabla u\|_{H^{1}(\mathbb{R}^{N})}^{2}-\tfrac 1{p+1}\| u\|_{L^{p+1}(\mathbb{R}^{N})}^{p+1}$ in relation to the value of the exponent $p+1\in (2, 2^{*}_{N})$.