On the Hénon-type Elliptic Systems Driven by the (p,1)-Laplacian Operator: Symmetry Breaking

In recent years, significant attention has been focused on issues related to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mn>1</mml:mn> </mml:math> -Laplacian operator, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . Motivated by these advancements and driven by the exploration of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> -Laplacian systems, we delve into the phenomenon of symmetry breaking in a Hénon-type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> -Laplacian system. Our primary finding establishes the existence of non-negative and non-radial solutions under specific parameter conditions. Initially, we examine the associated problem concerning the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> -Laplacian system, which is of independent interest. This is followed by an analysis of the sequence of solutions as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>q</mml:mi> </mml:math> approaches <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> asymptotically.