Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem
Global Components of Positive Bounded Variation Solutions of a One-Dimensional Indefinite Quasilinear Neumann Problem
Abstract This paper investigates the topological structure of the set of the positive solutions of the one-dimensional quasilinear indefinite Neumann problem <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo>{</m:mo><m:mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><m:mtr><m:mtd columnalign="left"><m:mrow><m:mrow><m:mrow><m:mo>-</m:mo><m:msup><m:mrow><m:mo maxsize="260%" minsize="260%">(</m:mo><m:mfrac><m:msup><m:mi>u</m:mi><m:mo>′</m:mo></m:msup><m:msqrt><m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:mmultiscripts><m:mi>u</m:mi><m:none/><m:mo>′</m:mo><m:none/><m:mn>2</m:mn></m:mmultiscripts></m:mrow></m:msqrt></m:mfrac><m:mo maxsize="260%" minsize="260%">)</m:mo></m:mrow><m:mo>′</m:mo></m:msup></m:mrow><m:mo>=</m:mo><m:mrow><m:mrow><m:mi>λ</m:mi><m:mo></m:mo><m:mi>a</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow><m:mo></m:mo><m:mi>f</m:mi><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>u</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo separator="true"> </m:mo><m:mrow><m:mtext>in </m:mtext><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mn>1</m:mn><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow></m:mtd><m:mtd/></m:mtr><m:mtr><m:mtd columnalign="left"><m:mrow><m:mrow><m:mrow><m:mrow><m:msup><m:mi>u</m:mi><m:mo>′</m:mo></m:msup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow><m:mo rspace="12.5pt">,</m:mo><m:mrow><m:mrow><m:msup><m:mi>u</m:mi><m:mo>′</m:mo></m:msup><m:mo></m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>1</m:mn><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo>=</m:mo><m:mn>0</m:mn></m:mrow></m:mrow><m:mo>,</m:mo></m:mrow></m:mtd><m:mtd/></m:mtr></m:mtable></m:mrow></m:math> \begin{dcases}-\Bigg{(}\frac{u^{\prime}}{\sqrt{1+{u^{\prime}}^{2}}}\Bigg{)}^{% \prime}=\lambda a(x)f(u)\quad\text{in }(0,1),\\ …