Local boundedness for solutions of a class of nonlinear elliptic systems

TL;DR

This paper establishes local boundedness of solutions for a class of nonlinear elliptic systems with p, q-growth conditions, using improved inequalities and iteration methods, under certain structural and exponent constraints.

Contribution

It introduces a componentwise coercivity condition ensuring regularity of solutions to nonlinear elliptic systems with p, q-growth, extending previous results to broader exponent ranges.

Findings

Solutions are locally bounded under certain structural conditions.

The method applies De Giorgi's iteration to systems with p, q-growth.

Results extend known bounds for specific p, q ranges.

Abstract

In this paper we are concerned with the regularity of solutions to a nonlinear elliptic system of $m$ equations in divergence form, satisfying $p$ growth from below and $q$ growth from above, with $p \leq q$; this case is known as $p, q$-growth conditions. Well known counterexamples, even in the simpler case $p=q$, show that solutions to systems may be singular; so, it is necessary to add suitable structure conditions on the system that force solutions to be regular. Here we obtain local boundedness of solutions under a componentwise coercivity condition. Our result is obtained by proving that each component $u^\alpha$ of the solution $u=(u^1,...,u^m)$ satisfies an improved Caccioppoli's inequality and we get the boundedness of $u^{\alpha}$ by applying De Giorgi's iteration method, provided the two exponents $p$ and $q$ are not too far apart. Let us remark that, in dimension $n=3$ and when $p=q$, our result works for $\frac{3}{2} < p < 3$, thus it complements the one of Bjorn whose technique allowed her to deal with $p \leq 2$ only. In the final section, we provide applications of our result.