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In this paper, we obtain gradient estimates of the positive solutions to weighted p-Laplacian type equations with a gradient-dependent nonlinearity of the form 0.1

$${\rm div }( \vert x \vert ^\sigma \vert \nabla u \vert ^{p-2}\nabla u) = \vert x \vert ^{-\tau }u^q \vert \nabla u \vert ^m\quad {\rm in}\;\Omega^*: = \Omega {\rm \setminus }\{ 0\} .$$

Here,

$\Omega \subseteq {\open R}^N$ denotes a domain containing the origin with

$N\ges 2$, whereas

$m,q\in [0,\infty )$,

$1<p\les N+\sigma $ and

$q>\max \{p-m-1,\sigma +\tau -1\}$. The main difficulty arises from the dependence of the right-hand side of (0.1) on x, u and

$ \vert \nabla u \vert $, without any upper bound restriction on the power m of

$ \vert \nabla u \vert $. Our proof of the gradient estimates is based on a two-step process relying on a modified version of the Bernstein's method. As a by-product, we extend the range of applicability of the Liouville-type results known for (0.1).

We prove pointwise gradient bounds for entire solutions of pde’s of the form

ℒu(x) = ψ(x, u(x), ∇u(x)),

where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.

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Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity

Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s

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2 results

Gradient estimates for nonlinear elliptic equations with a gradient-dependent nonlinearity

Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s