A non-existence result for the reinforced membrane problem

The paper deals with a shape optimization problem, known as reinforced membrane problem, where the reinforcement of the membrane over a subset E is modeled by a non-linear term in the governing PDE. The minimization of the energy (work done by the membrane), models finding the optimal shape of the reinforcement set E of prescribed volume. This problem does not always have a solution. We complement existing results about existence of solutions, by a non-existence result. In addition, we present an example for non-existence with a radially symmetric and radially decreasing force function, which rules out the possibility of a simple existence/non-existence alternative depending on the volume of the set E even in the radially symmetric situation.