Nontrivial solutions for a generalized poly-Laplacian system on finite graphs

Abstract We investigate the existence and multiplicity of solutions for a class of the generalized coupled system involving poly-Laplacian and the parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda on finite graphs. By using the Mountain pass lemma together with the cut-off technique, we obtain that system has at least a nontrivial weak solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left({u}_{\lambda },{v}_{\lambda }) for every large parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda when the nonlinear term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> F\left(x,u,v) satisfies superlinear growth conditions only in a neighborhood of origin point (0, 0). We also obtain a concrete form for the lower bound of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda and the trend of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left({u}_{\lambda },{v}_{\lambda }) with the change of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> \lambda . Moreover, by using a revised Clark’s theorem together with cut-off technique, we obtain that system has a sequence of solutions tending to 0 for every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:math> \lambda \gt 0 when the nonlinear term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>F</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> F\left(x,u,v) satisfies sublinear growth conditions only in a neighborhood of origin point (0, 0).