Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term

Abstract

Abstract We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable columnalign="left"> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mrow> <m:mo>-</m:mo> <m:mi>d</m:mi> <m:mi>i</m:mi> <m:mi>v</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>x</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>∇</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>γ</m:mi> </m:mrow> </m:mfrac> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>p</m:mi> </m:msup> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>|</m:mo> <m:mi>u</m:mi> <m:mo>|</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>θ</m:mi> </m:msup> </m:mrow> </m:mfrac> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi>i</m:mi> <m:mi>n</m:mi> <m:mi> </m:mi> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mi>o</m:mi> <m:mi>n</m:mi> <m:mi> </m:mi> <m:mo>∂</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:math> \left\{ {\matrix{ { - div\left( {b\left( x \right){{{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \over {\left( {1 + \left| u \right|} \right)\gamma }}} \right) + {{{{\left| {\nabla u} \right|}^p}} \over {{{\left| u \right|}^\theta }}} = f,} \hfill &amp; {in\,\Omega ,} \hfill \cr {u = 0,} \hfill &amp; {on\,\partial \Omega ,} \hfill \cr } } \right. swhere Ω is a bounded open subset in ℝ N , 1 ≤ θ &lt; 2, p &gt; 2 and γ &gt; 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f .