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• W2963181523 abstract "We derive explicit ground state solutions for several equations with the <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 in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R Superscript n> <mml:semantics> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding=application/x-tex>R^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, including (here <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi left-parenthesis z right-parenthesis equals z StartAbsoluteValue z EndAbsoluteValue Superscript p minus 2> <mml:semantics> <mml:mrow> <mml:mi>φ<!-- φ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>z</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mi>z</mml:mi> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>varphi (z)=z|z|^{p-2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than 1> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>p>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi left-parenthesis u prime left-parenthesis r right-parenthesis right-parenthesis Superscript prime Baseline plus StartFraction n minus 1 Over r EndFraction phi left-parenthesis u prime left-parenthesis r right-parenthesis right-parenthesis plus u Superscript upper M Baseline plus u Superscript upper Q Baseline equals 0 period> <mml:semantics> <mml:mrow> <mml:mi>φ<!-- φ --></mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>r</mml:mi> </mml:mfrac> <mml:mi>φ<!-- φ --></mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>M</mml:mi> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>Q</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=thinmathspace /> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>varphi left (u’(r)right )’ +frac {n-1}{r} varphi left (u’(r)right )+u^M+u^Q=0 ,.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> The constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper M greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>M>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is assumed to be below the critical power, while <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Q equals StartFraction upper M p minus p plus 1 Over p minus 1 EndFraction> <mml:semantics> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding=application/x-tex>Q=frac {M p-p+1}{p-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C. S. Lin and W.-M. Ni (1998). We also give the <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>-Laplace version of G. Bratu’s solution, connected to combustion theory. In another direction, we present a change of variables which removes the non-autonomous term <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=r Superscript alpha> <mml:semantics> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>r^{alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=phi left-parenthesis u prime left-parenthesis r right-parenthesis right-parenthesis Superscript prime Baseline plus StartFraction n minus 1 Over r EndFraction phi left-parenthesis u prime left-parenthesis r right-parenthesis right-parenthesis plus r Superscript alpha Baseline f left-parenthesis u right-parenthesis equals 0 comma> <mml:semantics> <mml:mrow> <mml:mi>φ<!-- φ --></mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>r</mml:mi> </mml:mfrac> <mml:mi>φ<!-- φ --></mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mspace width=thinmathspace /> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>varphi left (u’(r)right )’ +frac {n-1}{r} varphi left (u’(r)right )+r^{alpha } f(u)=0 ,,</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> while preserving the form of this equation. In particular, we study singular equations, when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, that occur often in applications. The Coulomb case <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=alpha equals negative 1> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>alpha =-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> turned out to give the critical power." @default.
• W2963181523 created "2019-07-30" @default.
• W2963181523 creator A5053724895 @default.
• W2963181523 date "2017-04-19" @default.
• W2963181523 modified "2023-09-24" @default.
• W2963181523 title "Explicit solutions and multiplicity results for some equations with the 𝑝-Laplacian" @default.
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• W2963181523 doi "https://doi.org/10.1090/qam/1471" @default.
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