Existence of solutions for singular quasilinear elliptic problems with dependence of the gradient

In this paper we establish existence of solutions to the following boundary value problem involving a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>p</a:mi> </a:math>-gradient term <b:math xmlns:b="http://www.w3.org/1998/Math/MathML" display="block"> <b:mstyle displaystyle="true" scriptlevel="0"> <b:mo>−</b:mo> <b:msub> <b:mi mathvariant="normal">Δ</b:mi> <b:mrow class="MJX-TeXAtom-ORD"> <b:mi>p</b:mi> </b:mrow> </b:msub> <b:mi>u</b:mi> <b:mo>+</b:mo> <b:mi>g</b:mi> <b:mo stretchy="false">(</b:mo> <b:mi>u</b:mi> <b:mo stretchy="false">)</b:mo> <b:mrow class="MJX-TeXAtom-ORD"> <b:mo stretchy="false">|</b:mo> </b:mrow> <b:mi mathvariant="normal">∇</b:mi> <b:mi>u</b:mi> <b:msup> <b:mrow class="MJX-TeXAtom-ORD"> <b:mo stretchy="false">|</b:mo> </b:mrow> <b:mi>p</b:mi> </b:msup> <b:mo>=</b:mo> <b:mi>λ</b:mi> <b:msup> <b:mi>u</b:mi> <b:mi>σ</b:mi> </b:msup> <b:mo>+</b:mo> <b:mi mathvariant="normal">Ψ</b:mi> <b:mo stretchy="false">(</b:mo> <b:mi>x</b:mi> <b:mo stretchy="false">)</b:mo> <b:mo>,</b:mo> <b:mspace width="1em"/> <b:mi>u</b:mi> <b:mo>&gt;</b:mo> <b:mn>0</b:mn> <b:mspace width="1em"/> <b:mstyle displaystyle="false" scriptlevel="0"> <b:mtext>in</b:mtext> </b:mstyle> <b:mtext> </b:mtext> <b:mi mathvariant="normal">Ω</b:mi> <b:mo>,</b:mo> <b:mspace width="1em"/> <b:mi>u</b:mi> <b:mo>=</b:mo> <b:mn>0</b:mn> <b:mspace width="1em"/> <b:mstyle displaystyle="false" scriptlevel="0"> <b:mtext>on</b:mtext> </b:mstyle> <b:mtext> </b:mtext> <b:mi mathvariant="normal">∂</b:mi> <b:mi mathvariant="normal">Ω</b:mi> <b:mo>,</b:mo> </b:mstyle> </b:math> where <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML"> <cb:msub> <cb:mi mathvariant="normal">Δ</cb:mi> <cb:mrow class="MJX-TeXAtom-ORD"> <cb:mi>p</cb:mi> </cb:mrow> </cb:msub> <cb:mo>:=</cb:mo> <cb:mi>div</cb:mi> <cb:mo>⁡</cb:mo> <cb:mo stretchy="false">(</cb:mo> <cb:mrow class="MJX-TeXAtom-ORD"> <cb:mo stretchy="false">|</cb:mo> </cb:mrow> <cb:mi mathvariant="normal">∇</cb:mi> <cb:mi>u</cb:mi> <cb:msup> <cb:mrow class="MJX-TeXAtom-ORD"> <cb:mo stretchy="false">|</cb:mo> </cb:mrow> <cb:mrow class="MJX-TeXAtom-ORD"> <cb:mi>p</cb:mi> <cb:mo>−</cb:mo> <cb:mn>2</cb:mn> </cb:mrow> </cb:msup> <cb:mi mathvariant="normal">∇</cb:mi> <cb:mi>u</cb:mi> <cb:mo stretchy="false">)</cb:mo> </cb:math> is <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML"> <ob:mi>p</ob:mi> </ob:math>-Laplacian operator, <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML"> <pb:mi mathvariant="normal">Ω</pb:mi> <pb:mo>⊂</pb:mo> <pb:msup> <pb:mrow class="MJX-TeXAtom-ORD"> <pb:mi mathvariant="double-struck">R</pb:mi> </pb:mrow> <pb:mi>N</pb:mi> </pb:msup> </pb:math> <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML"> <tb:mrow> <tb:mo>(</tb:mo> <tb:mrow> <tb:mi>N</tb:mi> <tb:mo>≥</tb:mo> <tb:mn>3</tb:mn> </tb:mrow> <tb:mo>)</tb:mo> </tb:mrow> </tb:math> is a bounded domain with smooth boundary, <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML"> <ub:mn>1</ub:mn> <ub:mo>&lt;</ub:mo> <ub:mi>p</ub:mi> <ub:mo>&lt;</ub:mo> <ub:mi>N</ub:mi> </ub:math>, <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML"> <vb:mn>0</vb:mn> <vb:mo>&lt;</vb:mo> <vb:mi>σ</vb:mi> <vb:mo>&lt;</vb:mo> <vb:msup> <vb:mi>p</vb:mi> <vb:mo>∗</vb:mo> </vb:msup> <vb:mo>−</vb:mo> <vb:mn>1</vb:mn> </vb:math> with <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML"> <wb:msup> <wb:mi>p</wb:mi> <wb:mo>∗</wb:mo> </wb:msup> <wb:mo>:=</wb:mo> <wb:mi>p</wb:mi> <wb:mi>N</wb:mi> <wb:mrow class="MJX-TeXAtom-ORD"> <wb:mo>/</wb:mo> </wb:mrow> <wb:mrow> <wb:mo>(</wb:mo> <wb:mrow> <wb:mi>N</wb:mi> <wb:mo>−</wb:mo> <wb:mi>p</wb:mi> </wb:mrow> <wb:mo>)</wb:mo> </wb:mrow> </wb:math>, <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML"> <yb:mi mathvariant="normal">Ψ</yb:mi> </yb:math> is a measurable function and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML"> <ac:mi>g</ac:mi> <ac:mo stretchy="false">(</ac:mo> <ac:mi>s</ac:mi> <ac:mo stretchy="false">)</ac:mo> <ac:mo>≥</ac:mo> <ac:mn>0</ac:mn> </ac:math> is a continuous function on the interval <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML"> <dc:mo stretchy="false">(</dc:mo> <dc:mn>0</dc:mn> <dc:mo>,</dc:mo> <dc:mo>+</dc:mo> <dc:mi mathvariant="normal">∞</dc:mi> <dc:mo stretchy="false">)</dc:mo> </dc:math> which may have a singularity at the origin, i.e.~<hc:math xmlns:hc="http://www.w3.org/1998/Math/MathML"> <hc:mi>g</hc:mi> <hc:mo stretchy="false">(</hc:mo> <hc:mi>s</hc:mi> <hc:mo stretchy="false">)</hc:mo> <hc:mo stretchy="false">→</hc:mo> <hc:mo>+</hc:mo> <hc:mi mathvariant="normal">∞</hc:mi> </hc:math> as <mc:math xmlns:mc="http://www.w3.org/1998/Math/MathML"> <mc:mi>s</mc:mi> <mc:mo stretchy="false">→</mc:mo> <mc:mn>0</mc:mn> </mc:math>. Using the topological degree theory, under certain assumptions on <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML"> <oc:mi mathvariant="normal">Ψ</oc:mi> </oc:math>, we prove the existence of a continuum of positive solutions.