Uniqueness of solutions for nonlinear elliptic problems with supercritical growth in ramified domains

We show that there exists only one solution (the trivial identically zero solution) for some nonlinear elliptic Dirichlet problems, involving the p-Laplacian operator and nonlinear terms with supercritical growth, in bounded contractible ramified domains, that is domains of [Formula: see text] with [Formula: see text], close to a prescribed subset of [Formula: see text], which is contractible in itself and consists of a finite number of smooth curves. In dimension [Formula: see text] we expect that this result might be extended to cover all the bounded contractible domains of [Formula: see text]. On the contrary, this extension is not possible in dimension [Formula: see text] because of some counterexamples concerning existence and multiplicity of nontrivial solutions in some contractible domains that may be even arbitrarily close to starshaped domains (where the well-known Pohozaev nonexistence result holds). However, also for [Formula: see text] our result allows us to prove nonexistence of nontrivial solutions in bounded contractible domains that may be very different from the starshaped ones and even arbitrarily close to some noncontractible domains where there exist many positive and nodal solutions.