Boundedness of solutions to quasilinear elliptic systems

Abstract This article deals with elliptic systems of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>−</m:mo> <m:munderover> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:munderover> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> <m:mfenced open="(" close=")"> <m:mrow> <m:munderover> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>β</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:munderover> <m:munderover> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>j</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:munderover> <m:msubsup> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>,</m:mo> <m:mi>j</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>f</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>α</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo>.</m:mo> </m:math> -\mathop{\sum }\limits_{i=1}^{n}\frac{\partial }{\partial {x}_{i}}\left(\mathop{\sum }\limits_{\beta =1}^{N}\mathop{\sum }\limits_{j=1}^{n}{a}_{i,j}^{\alpha ,\beta }\left(x,u\left(x))\frac{\partial {u}^{\beta }\left(x)}{\partial {x}_{j}}\right)={f}^{\alpha }\left(x),\hspace{1em}\alpha =1,\ldots ,N. Under ellipticity conditions of the diagonal coefficients and proportional conditions of the off-diagonal coefficients, we derive local and global boundedness results. Under ellipticity of all the coefficients and “butterfly” support of off-diagonal coefficients, we derive a global boundedness result. This article also considers regularizing effect of a lower-order term.