Existence and Convergence of Ground State Solutions for a ( <i>p</i> , <i>q</i> )-Laplacian System on Weighted Graphs

We investigate the existence of ground state solutions for a <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 with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> and potential wells on a weighted locally finite graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> . By making use of the method of Nehari manifold and the Lagrange multiplier rule, we prove that if the nonlinear term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> </mml:math> takes on the super- <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> -linear growth and the potential functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>a</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> satisfy some suitable conditions, then for any fixed parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>λ</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:math> , the system is provided with a ground state solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:math> . Additionally, we set up the convergence property of the solutions set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:math> when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>λ</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:math> .