Trudinger–Moser type inequalities with logarithmic weights in fractional dimensions

Abstract The purpose of this paper is two-fold. First, we derive sharp Trudinger–Moser inequalities with logarithmic weights in fractional dimensions: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mtext> </m:mtext> <m:munder> <m:mrow> <m:mi>sup</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:munderover accent="true" accentunder="false"> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:munderover> <m:mi>w</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>r</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>β</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>β</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:munder> <m:munderover accent="true" accentunder="false"> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mtext>e</m:mtext> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>θ</m:mi> <m:mo>,</m:mo> <m:mi>γ</m:mi> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mfenced close="|" open="|"> <m:mrow> <m:mi>u</m:mi> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mi>β</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mi>β</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mfenced> <m:mfenced close=")" open="("> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>γ</m:mi> </m:mrow> </m:mfenced> </m:mrow> </m:mfrac> </m:mrow> </m:msup> </m:mrow> </m:msup> <m:mi mathvariant="normal">d</m:mi> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>θ</m:mi> </m:mrow> </m:msub> <m:mo>&lt;</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> <m:mo>,</m:mo> </m:math> $$\,\underset{{\left(\underset{0}{\overset{1}{\int }}w\left(r\right){\left\vert {u}^{\prime }\left(r\right)\right\vert }^{\beta +2}\mathrm{d}{\lambda }_{\alpha }\right)}^{1/\left(\beta +2\right)}\le 1}{\mathrm{sup}}\underset{0}{\overset{1}{\int }}{\text{e}}^{{\mu }_{\alpha ,\theta ,\gamma }{\left\vert u\right\vert }^{\frac{\beta +2}{\left(\beta +1\right)\left(1-\gamma \right)}}}\mathrm{d}{\lambda }_{\theta }{&lt; }+\infty ,$$ where 0 ≤ γ &lt; 1, α = β + 1, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mi>α</m: