Local boundedness for solutions of a class of nonlinear elliptic systems

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 le 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 {le } 3, thus it complements the one of Bjorn whose technique allowed her to deal with p le 2 only. In the final section, we provide applications of our result.