Local boundedness for solutions of a class of nonlinear elliptic systems

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≤ 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 uα of the solution u= (u1,.. , um) satisfies an improved Caccioppoli’s inequality and we get the boundedness of uα 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 3/2