Sharp estimates of radial minimizers of 𝑝–Laplace equations

Abstract

We study semi-stable, radially symmetric and decreasing solutions u ∈ W 1 , p ( B 1 ) u\in W^{1,p}(B_1) of − Δ p u = g ( u ) -\Delta _p u=g(u) in B 1 ∖ { 0 } B_1\setminus \{ 0\} , where B 1 B_1 is the unit ball of R N \mathbb {R}^N , p > 1 p>1 , Δ p \Delta _p is the p − p- Laplace operator and g g is a general locally Lipschitz function. We establish sharp pointwise estimates for such solutions, which do not depend on the nonlinearity g g . By applying these results, sharp pointwise estimates are obtained for the extremal solution and its derivatives (up to order three) of the equation − Δ p u = λ f ( u ) -\Delta _p u=\lambda f(u) , posed in B 1 B_1 , with Dirichlet data u | ∂ B 1 = 0 u|_{\partial B_1}=0 , where the nonlinearity f f is an increasing C 1 C^1 function with f ( 0 ) > 0 f(0)>0 and lim t → + ∞ f ( t ) t p − 1 = + ∞ . \lim _{t\rightarrow +\infty }{\frac {f(t)}{t^{p-1}}}=+\infty .