Gradient Lagrangian Systems And Semilinear Pde

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 is an element of RN+1, including the Allen Cahn or the stationary Nonlinear Schrodinger case. In connection with this kind of problems we study some metric separation properties of sublevels of the functional V(u) = 1/2 parallel to del u parallel to(2)(H)(1)((R)(N)()) - 1/P+1 parallel to u parallel to(p+1)(Lp+1(RN)) in relation to the value of the exponent p + 1 is an element of (2, 2(N)*).