Local boundedness for solutions of a class of nonlinear elliptic systems

Abstract

AbstractIn 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 $$p \le q$$ p ≤ q ; this case is known as p, q-growth conditions. Well known counterexamples, even in the simpler case $$p=q$$ 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 $$u^\alpha $$ u α of the solution $$u=(u^1,...,u^m)$$ u = ( u 1 , . . . , u m ) satisfies an improved Caccioppoli’s inequality and we get the boundedness of $$u^{\alpha }$$ u α 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 $$n=3$$ n = 3 and when $$p=q$$ p = q , our result works for $$\frac{3}{2} < p {\le } 3$$ 3 2 < p ≤ 3 , thus it complements the one of Bjorn whose technique allowed her to deal with $$p \le 2$$ p ≤ 2 only. In the final section, we provide applications of our result.