Harnack inequality for quasilinear elliptic equations with (p,q) growth conditions and absorption lower order term
In this article we study the quasilinear elliptic equation with absorption lower term $$ -\hbox{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0, \quad u\geq 0. $$ Despite of the lack of comparison principle, we prove a priori estimate of Keller-Osserman type. Particularly, und...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-04-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/91/abstr.html |
| Summary: | In this article we study the quasilinear elliptic equation with absorption
lower term
$$
-\hbox{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0,
\quad u\geq 0.
$$
Despite of the lack of comparison principle, we prove a priori
estimate of Keller-Osserman type. Particularly, under some natural
assumptions on the functions g,f for nonnegative solutions we
prove an estimate of the form
$$
\int_0^{u(x)} f(s)\,ds\leq c\frac{u(x)}{r}g\big(\frac{u(x)}{r}\big),\quad
x\in\Omega, B_{8r}(x)\subset\Omega,
$$
with constant c, independent on u(x). Using this estimate we
give a simple proof of the Harnack inequality. |
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| ISSN: | 1072-6691 |