We show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ${{\mathbb{R}}^{n}}$ has infinitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as ${{\left| s \right|}^{p-1}}s$ for $\left| s \right|$ large with $1\,<\,p\,<\,\left( n+5 \right)/\left( n-3 \right)$.

Motivated by a classical work of Erdős we give rather precise necessary and sufficient growth conditions on the nonlinearity in a semilinear wave equation in order to have global existence for all initial data. Then we improve some former exact controllability theorems of Imanuvilov and Zuazua.

We study here an optimal control problem for a semilinear elliptic equation with an exponential nonlinearity, such that we cannotexpect to have a solution of the state equation for any given control. We then have to speak of pairs (control, state). After havingdefined a suitable functional class in which we look for solutions, we prove existence of an optimal pair for a large class of costfunctions using a non standard compactness argument. Then, we derive a first order optimality system assuming the optimal pair isslightly more regular.