Local boundedness for solutions of a class of nonlinear elliptic systems
Giovanni Cupini, Francesco Leonetti, Elvira Mascolo
Abstract
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 $$p \le q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> ; this case is known as p , q -growth conditions. Well known counterexamples, even in the simpler case $$p=q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> , 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 $$u^\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:math> of the solution $$u=(u^1,...,u^m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> satisfies an improved Caccioppoli’s inequality and we get the boundedness of $$u^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:math> 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 $$n=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and when $$p=q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </mml:math> , our result works for $$\frac{3}{2} < p {\le } 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>3</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo><</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , thus it complements the one of Bjorn whose technique allowed her to deal with $$p \le 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> only. In the final section, we provide applications of our result.