Existence and regularity of solutions for a singular anisotropic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mo stretchy="false">(</a:mo> <a:mi>p</a:mi> <a:mo>,</a:mo> <a:mi>q</a:mi> <a:mo stretchy="false">)</a:mo></a:math>-Laplacian wit

In this paper, we investigate the existence and regularity of positive solutions for certain singular problems that involve an anisotropic <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mo stretchy="false">(</a:mo> <a:mi>p</a:mi> <a:mo>,</a:mo> <a:mi>q</a:mi> <a:mo stretchy="false">)</a:mo> </a:math>-Laplacian-type operator and a singular term with a variable exponent, under zero Dirichlet boundary conditions on <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi mathvariant="normal">∂</d:mi> <d:mi mathvariant="normal">Ω</d:mi> </d:math>. The main equation we analyze is <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="block"> <g:mo>−</g:mo> <g:munderover> <g:mo movablelimits="false">∑</g:mo> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> <g:mo>=</g:mo> <g:mn>1</g:mn> </g:mrow> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>N</g:mi> </g:mrow> </g:munderover> <g:msub> <g:mi mathvariant="normal">∂</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mrow> <g:mo>(</g:mo> <g:mrow> <g:msup> <g:mrow class="MJX-TeXAtom-ORD"> <g:mrow class="MJX-TeXAtom-ORD"> <g:mo stretchy="false">|</g:mo> </g:mrow> <g:msub> <g:mi mathvariant="normal">∂</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mi>u</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo fence="false" stretchy="false">|</g:mo> </g:mrow> <g:mrow class="MJX-TeXAtom-ORD"> <g:msub> <g:mi>p</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mo>−</g:mo> <g:mn>2</g:mn> </g:mrow> </g:msup> <g:msub> <g:mi mathvariant="normal">∂</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mi>u</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:mo stretchy="false">)</g:mo> </g:mrow> <g:mo>)</g:mo> </g:mrow> <g:mo>−</g:mo> <g:munderover> <g:mo movablelimits="false">∑</g:mo> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> <g:mo>=</g:mo> <g:mn>1</g:mn> </g:mrow> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>N</g:mi> </g:mrow> </g:munderover> <g:msub> <g:mi mathvariant="normal">∂</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mrow> <g:mo>(</g:mo> <g:mrow> <g:msup> <g:mrow class="MJX-TeXAtom-ORD"> <g:mrow class="MJX-TeXAtom-ORD"> <g:mo stretchy="false">|</g:mo> </g:mrow> <g:msub> <g:mi mathvariant="normal">∂</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mi>u</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo fence="false" stretchy="false">|</g:mo> </g:mrow> <g:mrow class="MJX-TeXAtom-ORD"> <g:msub> <g:mi>q</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mo>−</g:mo> <g:mn>2</g:mn> </g:mrow> </g:msup> <g:msub> <g:mi mathvariant="normal">∂</g:mi> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>i</g:mi> </g:mrow> </g:msub> <g:mi>u</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:mo stretchy="false">)</g:mo> </g:mrow> <g:mo>)</g:mo> </g:mrow> <g:mo>=</g:mo> <g:mfrac> <g:mrow> <g:mi>f</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:mo stretchy="false">)</g:mo> </g:mrow> <g:mrow> <g:mi>u</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:msup> <g:mo stretchy="false">)</g:mo> <g:mrow class="MJX-TeXAtom-ORD"> <g:mi>γ</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>x</g:mi> <g:mo stretchy="false">)</g:mo> </g:mrow> </g:msup> </g:mrow> </g:mfrac> <g:mspace width="1em"/> <g:mtext>in </g:mtext> <g:mspace width="thickmathspace"/> <g:mi mathvariant="normal">Ω</g:mi> <g:mo>,</g:mo> </g:math> where <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML"> <gc:mi mathvariant="normal">Ω</gc:mi> </gc:math> is a bounded, regular domain in <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML"> <ic:msup> <ic:mrow class="MJX-TeXAtom-ORD"> <ic:mi mathvariant="double-struck">R</ic:mi> </ic:mrow> <ic:mi>N</ic:mi> </ic:msup> </ic:math>, <lc:math xmlns:lc="http://www.w3.org/1998/Math/MathML"> <lc:mi>f</lc:mi> </lc:math> is a positive function belonging to a specific Lebesgue space, and <mc:math xmlns:mc="http://www.w3.org/1998/Math/MathML"> <mc:mi>γ</mc:mi> <mc:mo stretchy="false">(</mc:mo> <mc:mi>x</mc:mi> <mc:mo stretchy="false">)</mc:mo> </mc:math> is a positive continuous function on <pc:math xmlns:pc="http://www.w3.org/1998/Math/MathML"> <pc:mover> <pc:mi mathvariant="normal">Ω</pc:mi> <pc:mo accent="false">¯</pc:mo> </pc:mover> </pc:math>. In our study, we do not make comparisons between <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML"> <sc:msub> <sc:mi>p</sc:mi> <sc:mi>i</sc:mi> </sc:msub> </sc:math> and <tc:math xmlns:tc="http://www.w3.org/1998/Math/MathML"> <tc:msub> <tc:mi>q</tc:mi> <tc:mi>i</tc:mi> </tc:msub> </tc:math>, and as a result, we show that the solution belongs to either <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML"> <uc:msubsup> <uc:mi>W</uc:mi> <uc:mn>0</uc:mn> <uc:mrow class="MJX-TeXAtom-ORD"> <uc:mn>1</uc:mn> <uc:mo>,</uc:mo> <uc:mrow class="MJX-TeXAtom-ORD"> <uc:mover> <uc:mi>p</uc:mi> <uc:mo stretchy="false">→</uc:mo> </uc:mover> </uc:mrow> </uc:mrow> </uc:msubsup> <uc:mo stretchy="false">(</uc:mo> <uc:mi mathvariant="normal">Ω</uc:mi> <uc:mo stretchy="false">)</uc:mo> <uc:mo>∩</uc:mo> <uc:msubsup> <uc:mi>W</uc:mi> <uc:mn>0</uc:mn> <uc:mrow class="MJX-TeXAtom-ORD"> <uc:mn>1</uc:mn> <uc:mo>,</uc:mo> <uc:mrow class="MJX-TeXAtom-ORD"> <uc:mover> <uc:mi>q</uc:mi> <uc:mo stretchy="false">→</uc:mo> </uc:mover> </uc:mrow> </uc:mrow> </uc:msubsup> <uc:mo stretchy="false">(</uc:mo> <uc:mi mathvariant="normal">Ω</uc:mi> <uc:mo stretchy="false">)</uc:mo> </uc:math> or <hd:math xmlns:hd="http://www.w3.org/1998/Math/MathML"> <hd:msubsup> <hd:mi>W</hd:mi> <hd:mrow class="MJX-TeXAtom-ORD"> <hd:mi>l</hd:mi> <hd:mi>o</hd:mi> <hd:mi>c</hd:mi> </hd:mrow> <hd:mrow class="MJX-TeXAtom-ORD"> <hd:mn>1</hd:mn> <hd:mo>,</hd:mo> <hd:mrow class="MJX-TeXAtom-ORD"> <hd:mover> <hd:mi>p</hd:mi> <hd:mo stretchy="false">→</hd:mo> </hd:mover> </hd:mrow> </hd:mrow> </hd:msubsup> <hd:mo stretchy="false">(</hd:mo> <hd:mi mathvariant="normal">Ω</hd:mi> <hd:mo stretchy="false">)</hd:mo> <hd:mo>∩</hd:mo> <hd:msubsup> <hd:mi>W</hd:mi> <hd:mrow class="MJX-TeXAtom-ORD"> <hd:mi>l</hd:mi> <hd:mi>o</hd:mi> <hd:mi>c</hd:mi> </hd:mrow> <hd:mrow class="MJX-TeXAtom-ORD"> <hd:mn>1</hd:mn> <hd:mo>,</hd:mo> <hd:mrow class="MJX-TeXAtom-ORD"> <hd:mover> <hd:mi>q</hd:mi> <hd:mo stretchy="false">→</hd:mo> </hd:mover> </hd:mrow> </hd:mrow> </hd:msubsup> <hd:mo stretchy="false">(</hd:mo> <hd:mi mathvariant="normal">Ω</hd:mi> <hd:mo stretchy="false">)</hd:mo> </hd:math> depending on the summability of <wd:math xmlns:wd="http://www.w3.org/1998/Math/MathML"> <wd:mi>f</wd:mi> <wd:mo stretchy="false">(</wd:mo> <wd:mi>x</wd:mi> <wd:mo stretchy="false">)</wd:mo> </wd:math> and the values of <zd:math xmlns:zd="http://www.w3.org/1998/Math/MathML"> <zd:mi>γ</zd:mi> <zd:mo stretchy="false">(</zd:mo> <zd:mi>x</zd:mi> <zd:mo stretchy="false">)</zd:mo> </zd:math>. The results are achieved using approximation techniques that include truncation, comparison, and variational methods.