Quasilinear elliptic equations with positive exponent on the gradient

Abstract

We study the existence and nonexistence of positive, spherically symmetric solutions of a quasilinear elliptic equation (1.1) involving p-Laplace operator, with an arbitrary positive growth rate $e_0$ on the gradient on the right-hand side. We show that $e_0 = p − 1$ is the critical exponent: for $e_0 p − 1$ we have existence-nonexistence splitting of the coefficients $\tilde f_0$ and $\tilde g_0$. The elliptic problem is studied by relating it to the corresponding singular ODE of the first order. We give sufficient conditions for a strong radial solution to be the weak solution. We also examined when -solutions of (1.1), defined in Definition 2.3, are weak solutions. We found conditions under which strong solutions are weak solutions in the critical case of $e_0 = p − 1$.