Computation and Analysis of Global Solution Curves for Super-Critical Equations

We study analytical and computational aspects for Dirichlet problem on the unit ball B: |x|<1 in Rn, modeled on the equation Δu+λ(up+uq)=0, in B, u=0 on ∂B, with a positive parameter λ, and 1 nn+2. This was already observed by I. Flores [Citation5], who proved the existence of infinitely many ground state solutions. We study properties of infinitely many solution curves of Equation(1.1)1.1 Δu+λ(up+uq)=0, in B, u=0 on ∂B,1.1 that are separated by these ground state solutions. We also study singular solutions (where u(0)=∞), and again the Lin-Ni equation plays a special role. Super-critical equations are very challenging computationally: solutions exist only for very large λ, and curves of positive solutions make turns at very large values of u(0)=∥u∥L∞. We overcome these difficulties by developing new results on singular solutions, and by using some delicate capabilities of Mathematica software.