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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2024-11-01
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| author | Dandan Yang Zhenyu Bai Chuanzhi Bai |
| author_facet | Dandan Yang Zhenyu Bai Chuanzhi Bai |
| author_sort | Dandan Yang |
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| description | 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. |
| format | Article |
| id | doaj-art-38a98f476e2d45a6bd4fce8d3840b787 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-38a98f476e2d45a6bd4fce8d3840b7872024-11-26T17:50:49ZengMDPI AGAxioms2075-16802024-11-01131176210.3390/axioms13110762Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice GraphsDandan Yang0Zhenyu Bai1Chuanzhi Bai2School of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, ChinaSchool of Mathematics and Statistics, Guizhou University, Guiyang 550025, ChinaSchool of Mathematics and Statistics, Huaiyin Normal University, Huaian 223300, ChinaIn 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.https://www.mdpi.com/2075-1680/13/11/762Choquard equation(<i>p</i>, <i>q</i>)-Laplacianmountain pass theoremlattice graphs |
| spellingShingle | Dandan Yang Zhenyu Bai Chuanzhi Bai Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs Axioms Choquard equation (<i>p</i>, <i>q</i>)-Laplacian mountain pass theorem lattice graphs |
| title | Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs |
| title_full | Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs |
| title_fullStr | Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs |
| title_full_unstemmed | Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs |
| title_short | Existence of Solutions for Nonlinear Choquard Equations with (<i>p</i>, <i>q</i>)-Laplacian on Finite Weighted Lattice Graphs |
| title_sort | existence of solutions for nonlinear choquard equations with i p i i q i laplacian on finite weighted lattice graphs |
| topic | Choquard equation (<i>p</i>, <i>q</i>)-Laplacian mountain pass theorem lattice graphs |
| url | https://www.mdpi.com/2075-1680/13/11/762 |
| work_keys_str_mv | AT dandanyang existenceofsolutionsfornonlinearchoquardequationswithipiiqilaplacianonfiniteweightedlatticegraphs AT zhenyubai existenceofsolutionsfornonlinearchoquardequationswithipiiqilaplacianonfiniteweightedlatticegraphs AT chuanzhibai existenceofsolutionsfornonlinearchoquardequationswithipiiqilaplacianonfiniteweightedlatticegraphs |