Existence and uniqueness theorems for some semi-linear equations on locally finite graphs
Andrea Pinamonti, Giorgio Stefani
Abstract
We study some semi-linear equations for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis m comma p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(m,p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacian operator on locally finite weighted graphs. We prove existence of weak solutions for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m element-of double-struck upper N"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">m\in \mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-parenthesis 1 comma plus normal infinity right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal"> ∞ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p\in (1,+\infty )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via a variational method already known in the literature by exploiting the continuity properties of the energy functionals involved. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we also establish a uniqueness result in the spirit of the Brezis–Strauss Theorem. We finally provide some applications of our main results by dealing with some Yamabe-type and Kazdan–Warner-type equations on locally finite weighted graphs.