The Dirichlet problem for the logarithmic 𝑝-Laplacian

We introduce and study the logarithmic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal upper Delta Sub Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{\Delta _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which emerges from the formal derivative of the fractional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacian <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis minus normal upper Delta Subscript p Baseline right-parenthesis Superscript s"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mo> − </mml:mo> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>s</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">(-\Delta _p)^s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s equals 0"> <mml:semantics> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">s=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator (see H. Chen and T. Weth [Comm. Partial Differential Equations 44 (2019), pp. 1100–1139]). We present a variational framework to study the Dirichlet problems involving the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal upper Delta Sub Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{\Delta _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacian and the logarithmic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal upper Delta Sub Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{\Delta _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We discuss maximum and comparison principles for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal upper Delta Sub Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{\Delta _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal upper Delta Sub Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L_{\Delta _p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In addition, we prove that the first Dirichlet eigenfunction of <inline-formula content-type="math/mathml"> <mml:math