Nodal Solutions for Supercritical Laplace Equations

In this paper we study radial solutions for the following equation $$ Delta u(x)+f(u(x),|x|)=0,$$ where $x inmathbb{R}^n$, $n>2$, $f$ is subcritical for $r$ small and $u$ large and supercritical for $r$ large and $u$ small, with respect to the Sobolev critical exponent $2^*=rac{2n}{n-2}$. The solutions are classified and characterized by their asymptotic behaviour and nodal properties. In an appropriate super-linear setting, we give an asymptotic condition sufficient to guarantee the existence of at least one ground state with fast decay with exactly $j$ zeroes for any $j ge 0$. Under the same assumptions, we also find uncountably many ground states with slow decay, singular ground states with fast decay and singular ground states with slow decay, all of them with exactly $j$ zeroes. Our approach, based on Fowler transformation and invariant manifold theory, enables us to deal with a wide family of potentials allowing spatial inhomogeneity and a quite general dependence on $u$. In particular, for the Matukuma-type potential, we show a kind of structural stability.