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The following semi-linear elliptic equations involving Hardy–Sobolev critical exponents <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><mo>Δ</mo><mi>u</mi><mo>−</mo><mi>μ</mi><mfrac><mi>u</mi><msup><mfenced open="|" close="|"><mi>x</mi></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><msup><mfenced open="|" close="|"><mi>u</mi></mfenced><mrow><msup><mn>2</mn><mo>*</mo></msup><mfenced open="(" close=")"><mi>s</mi></mfenced><mo>−</mo><mn>2</mn></mrow></msup><msup><mfenced open="|" close="|"><mi>x</mi></mfenced><mi>s</mi></msup></mfrac><mi>u</mi><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mo>,</mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">Ω</mi><mo>∖</mo><mfenced open="{" close="}"><mn>0</mn></mfenced><mo>,</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>∈</mo><mo>∂</mo><mi mathvariant="normal">Ω</mi></mrow></semantics></math></inline-formula> have been investigated, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Ω</mi></semantics></math></inline-formula> is an open-bounded domain in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup><mfenced separators="" open="(" close=")"><mi>N</mi><mo>≥</mo><mn>3</mn></mfenced></mrow></semantics></math></inline-formula>, with a smooth boundary <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>∂</mo><mi mathvariant="normal">Ω</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>∈</mo><mi mathvariant="normal">Ω</mi><mo>,</mo><mspace width="0.166667em"></mspace><mn>0</mn><mo>≤</mo><mi>μ</mi><mo><</mo><mover accent="true"><mi>μ</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><mo>=</mo><msup><mfenced separators="" open="(" close=")"><mfrac><mrow><mi>N</mi><mo>−</mo><mn>2</mn></mrow><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>,</mo><mspace width="0.277778em"></mspace><mn>0</mn><mo>≤</mo><mi>s</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mo>*</mo></msup><mfenced open="(" close=")"><mi>s</mi></mfenced><mo>=</mo><mn>2</mn><mfenced separators="" open="(" close=")"><mi>N</mi><mo>−</mo><mi>s</mi></mfenced><mo>/</mo><mfenced separators="" open="(" close=")"><mi>N</mi><mo>−</mo><mn>2</mn></mfenced></mrow></semantics></math></inline-formula> is the Hardy–Sobolev critical exponent. This problem comes from the study of standing waves in the anisotropic Schrödinger equation; it is very important in the fields of hydrodynamics, glaciology, quantum field theory, and statistical mechanics. Under some deterministic conditions on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi></mrow></semantics></math></inline-formula>, by a detailed estimation of the extremum function and using mountain pass lemma with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced separators="" open="(" close=")"><mi>P</mi><mi>S</mi></mfenced><mi>c</mi></msub></semantics></math></inline-formula> conditions, we obtained that: (a) If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>μ</mi><mo>≤</mo><mover><mi>μ</mi><mo>¯</mo></mover><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo><</mo><msub><mi>λ</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>μ</mi></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> then the above problem has at least a positive solution in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mrow><mn>0</mn></mrow><mn>1</mn></msubsup><mfenced open="(" close=")"><mi mathvariant="normal">Ω</mi></mfenced></mrow></semantics></math></inline-formula>; (b) If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>μ</mi><mo>¯</mo></mover><mo>−</mo><mn>1</mn><mo><</mo><mi>μ</mi><mo><</mo><mover><mi>μ</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula>, then when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>λ</mi><mo>*</mo></msub><mfenced open="(" close=")"><mi>μ</mi></mfenced><mo><</mo><mi>λ</mi><mo><</mo><msub><mi>λ</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>μ</mi></mfenced></mrow></semantics></math></inline-formula>, the above problem has at least a positive solution in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mrow><mn>0</mn></mrow><mn>1</mn></msubsup><mfenced open="(" close=")"><mi mathvariant="normal">Ω</mi></mfenced></mrow></semantics></math></inline-formula>; (c) if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi>μ</mi><mo>¯</mo></mover><mo>−</mo><mn>1</mn><mo><</mo><mi>μ</mi><mo><</mo><mover><mi>μ</mi><mo>¯</mo></mover></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="normal">Ω</mi><mo>=</mo><mi>B</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, then the above problem has no positive solution for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>≤</mo><msub><mi>λ</mi><mo>*</mo></msub><mfenced open="(" close=")"><mi>μ</mi></mfenced><mo>.</mo></mrow></semantics></math></inline-formula> These results are extensions of E. Jannelli’s research (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>λ</mi><mi>u</mi></mrow></semantics></math></inline-formula>).