Uniqueness of solutions to singular p-Laplacian equations with subcritical nonlinearity

Abstract

We present a geometric approach to the study of quasilinear elliptic p-Laplacian problems on a ball in ℝn${\mathbb{R}^{n}}$ using techniques from dynamical systems. These techniques include a study of the invariant manifolds that arise from the union of the solutions to the elliptic PDE in phase space, as well as variational computations on two vector fields tangent to the invariant manifolds. We show that for a certain class of nonlinearities f with subcritical growth relative to the Sobolev critical exponent p*${p^{*}}$, there can be at most one such solution satisfying Δp⁢u+f⁢(u)=0${\Delta_{p}u+f(u)=0}$ on a ball with Dirichlet boundary conditions.

Keywords

• uniqueness
• quasilinear elliptic equations
• emden–fowler transformation
• invariant manifold
• 34c45
• 35j75
• 35j92
• 37b55