Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs

Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs

In this paper, we consider the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow><...

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Bibliographic Details
Main Authors:	Dandan Yang, Zhenyu Bai, Chuanzhi Bai
Format:	Article
Language:	English
Published:	MDPI AG 2024-11-01
Series:	Axioms
Subjects:	Choquard equation (<i>p</i>, <i>q</i>)-Laplacian mountain pass theorem lattice graphs
Online Access:	https://www.mdpi.com/2075-1680/13/11/762
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Summary:	In this paper, we consider the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Laplacian Choquard equation on a finite weighted lattice graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><msup><mi>K</mi><mi>N</mi></msup><mo>,</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>ω</mi><mo>)</mo></mrow></semantics></math></inline-formula>, namely for any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>q</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><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><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mi>N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>−</mo><msub><mo>Δ</mo><mi>p</mi></msub><mi>u</mi><mo>−</mo><msub><mo>Δ</mo><mi>q</mi></msub><mi>u</mi><mo>+</mo><msup><mrow><mi>V</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>(</mo><mo>\|</mo><mi>u</mi><mo>\|</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><msup><mrow><mo>\|</mo><mi>u</mi><mo>\|</mo></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mrow><mi>u</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>=</mo><mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>y</mi><mo>∈</mo><msup><mi>K</mi><mi>N</mi></msup><mo>,</mo><mi>y</mi><mo>≠</mo><mi>x</mi></mrow></munder></mstyle><mstyle scriptlevel="0" displaystyle="true"><mfrac><msup><mrow><mo>\|</mo><mi>u</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>\|</mo></mrow><mi>r</mi></msup><mrow><mi>d</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mrow><mi>N</mi><mo>−</mo><mi>α</mi></mrow></msup></mrow></mfrac></mstyle></mfenced><msup><mrow><mo>\|</mo><mi>u</mi><mo>\|</mo></mrow><mrow><mi>r</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Δ</mo><mi>ν</mi></msub></semantics></math></inline-formula> is the discrete <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ν</mi></semantics></math></inline-formula>-Laplacian on graphs, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ν</mi><mo>∈</mo><mo>{</mo><mi>p</mi><mo>.</mo><mi>q</mi><mo>}</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> is a positive function. Under some suitable conditions on <i>r</i>, we prove that the above equation has both a mountain pass solution and ground state solution. Our research relies on the mountain pass theorem and the method of the Nehari manifold. The results obtained in this paper are extensions of some known studies.
ISSN:	2075-1680