Low energy nodal solutions to the Yamabe equation

Supercritical elliptic problems on the round sphere and nodal solutions to the Yamabe problem in projective spaces

“…Our Theorem improves the existence result stated by Henry in [27], giving an infinite number of distinct solutions instead of one. It also extends the multiplicity result in [22] to the subcritical and supercritical exponents. However, this last result gives a better description of the nodal set of the solutions.…”

“…If u denotes a sign-changing smooth function defined on a Riemannian manifold, we define the nodal and the critical sets of u to be the sets {u = 0} and {∇u = 0}, respectively. We state the main result of this paper, which generalizes Theorem 1.3 in [22].…”

“…However, this last result gives a better description of the nodal set of the solutions. We strongly believe that a refinement of our methods may give a prescribed number of connected components for the nodal sets of the sign-changing solutions to problem (1.1), as in Theorem 1.2 in [22].…”

“…Here g 0 denotes the round metric and we will assume from now on that λ > 0 is constant. When p = p n , (1.2) is a renormalization of the Yamabe problem (1.1) and this kind of solutions have been studied in [13,15,19,20,37] and more recently in [22,31,40]. The slightly subcritical case has been studied in [42], where the authors obtained multiplicity of nodal solutions blowing-up at points, while the general subcritical case has been studied in [28] and in [8].…”

“…One of the few results known by the authors is given in [27], where the existence of at least one signchanging solution to the supercritical problem was settled. In this direction we will follow and generalize the ideas introduced in [22] to obtain an infinite number of nodal solutions to the supercritical and subcritical problem (1.2), having as level and critical sets isoparametric hypersurfaces and its focal submanifolds.…”