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• W4312588749 abstract "Abstract In this work, we study the weighted Kirchhoff problem <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mfenced open={ close=> <m:mrow> <m:mtable displaystyle=true> <m:mtr> <m:mtd columnalign=left> <m:mo>−</m:mo> <m:mi>g</m:mi> <m:mfenced open=( close=)> <m:mrow> <m:munder> <m:mrow> <m:mrow> <m:mstyle displaystyle=true> <m:mo>∫</m:mo> </m:mstyle> </m:mrow> </m:mrow> <m:mrow> <m:mi>B</m:mi> </m:mrow> </m:munder> <m:mi>σ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mspace width=-0.25em /> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mi mathvariant=normal>d</m:mi> <m:mi>x</m:mi> </m:mrow> </m:mfenced> <m:mi mathvariant=normal>div</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>σ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mspace width=-0.25em /> <m:msup> <m:mrow> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <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:mrow> <m:mo>)</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=left> <m:mspace width=0.1em /> <m:mtext>in</m:mtext> <m:mspace width=0.1em /> <m:mspace width=0.33em /> <m:mi>B</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=left> <m:mi>u</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mtd> <m:mtd columnalign=left> <m:mspace width=0.1em /> <m:mtext>in</m:mtext> <m:mspace width=0.1em /> <m:mspace width=0.33em /> <m:mi>B</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=left> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mtd> <m:mtd columnalign=left> <m:mspace width=0.1em /> <m:mtext>on</m:mtext> <m:mspace width=0.1em /> <m:mspace width=0.33em /> <m:mo>∂</m:mo> <m:mi>B</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> left{begin{array}{ll}-gleft(mathop{displaystyle int }limits_{B}sigma left(x)| nabla uhspace{-0.25em}{| }^{N}{rm{d}}xright){rm{div}}left(sigma left(x)| nabla uhspace{-0.25em}{| }^{N-2}nabla u)=fleft(x,u)& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}B, ugt 0& hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}B, u=0& hspace{0.1em}text{on}hspace{0.1em}hspace{0.33em}partial B,end{array}right. where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>B</m:mi> </m:math> B is the unit ball of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi mathvariant=double-struck>R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:math> {{mathbb{R}}}^{N} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>σ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mfenced open=( close=)> <m:mrow> <m:mi>log</m:mi> <m:mfenced open=( close=)> <m:mrow> <m:mfrac> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mo>∣</m:mo> <m:mspace width=-0.25em /> <m:mi>x</m:mi> <m:mspace width=-0.25em /> <m:mo>∣</m:mo> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:mrow> </m:mfenced> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> sigma left(x)={left(log left(frac{e}{| x| }right)right)}^{N-1} , the singular logarithm weight in the Trudinger-Moser embedding, and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>g</m:mi> </m:math> g is a continuous positive function on <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:msup> <m:mrow> <m:mi mathvariant=double-struck>R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:math> {{mathbb{R}}}^{+} . The nonlinearity is critical or subcritical growth in view of Trudinger-Moser inequalities. We first obtain the existence of a solution in the subcritical exponential growth case with positive energy by using minimax techniques combined with the Trudinger-Moser inequality. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth, and we stress its importance to check the compactness level." @default.
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• W4312588749 date "2022-01-01" @default.
• W4312588749 modified "2023-10-18" @default.
• W4312588749 title "On a weighted elliptic equation of N-Kirchhoff type with double exponential growth" @default.
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• W4312588749 doi "https://doi.org/10.1515/dema-2022-0156" @default.
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