Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
Abstract This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"><m:mfenced open="{" close=""><m:mrow><m:mtable displaystyle="true" columnalign="left"><m:mtr columnalign="left"><m:mtd columnalign="left"><m:mrow><m:mo>−</m:mo><m:mtext>div</m:mtext><m:mrow><m:mfenced open="(" close=")"><m:mrow><m:mfrac><m:mrow><m:mo>∇</m:mo><m:mi>v</m:mi></m:mrow><m:mrow><m:msqrt><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:mo>|</m:mo><m:mo>∇</m:mo><m:mi>v</m:mi><m:msup><m:mrow><m:mo>|</m:mo></m:mrow><m:mn>2</m:mn></m:msup></m:mrow></m:msqrt></m:mrow></m:mfrac></m:mrow></m:mfenced></m:mrow><m:mo>=</m:mo><m:mi>f</m:mi><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo>,</m:mo><m:mi>v</m:mi><m:mo>,</m:mo><m:mo>∇</m:mo><m:mi>v</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mtd><m:mtd columnalign="left"><m:mrow><m:mtext>in</m:mtext><m:mspace width=".5em" /><m:mtext>Ω</m:mtext><m:mo>,</m:mo></m:mrow></m:mtd></m:mtr><m:mtr columnalign="left"><m:mtd columnalign="left"><m:mrow><m:msub><m:mrow><m:mi>a</m:mi></m:mrow><m:mrow><m:mn>0</m:mn></m:mrow></m:msub><m:mi>v</m:mi><m:mo>+</m:mo><m:msub><m:mrow><m:mi>a</m:mi></m:mrow><m:mrow><m:mn>1</m:mn></m:mrow></m:msub><m:mstyle displaystyle="false"><m:mfrac><m:mrow><m:mi>∂</m:mi><m:mi>v</m:mi></m:mrow><m:mrow><m:mi>∂</m:mi><m:mi>ν</m:mi></m:mrow></m:mfrac></m:mstyle><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow></m:mtd><m:mtd columnalign="left"><m:mrow><m:mtext>on</m:mtext><m:mspace width=".5em" /><m:mi>∂</m:mi><m:mtext>Ω</m:mtext><m:mo>,</m:mo></m:mrow></m:mtd></m:mtr></m:mtable></m:mrow></m:mfenced></m:math> \left\{\begin{array}{ll}-\text{div}\left(\frac{\nabla v}{\sqrt{1+|\nabla v{|}^{2}}}\right)=f(x,v,\nabla v)& \text{in}\hspace{.5em}\text{&#x03A9;},\\ {a}_{0}v+{a}_{1}\tfrac{\partial v}{\partial \nu …