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 ℝn with n ≥ 2, close to a prescribed subset of n, which is contractible in itself and consists of a finite number of smooth curves. In dimension n = 2 we expect that this result might be extended to cover all the bounded contractible domains of ℝ2. On the contrary, this extension is not possible in dimension n ≥ 3 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 n ≥ 3 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.