Eigenvalue problem with indefinite weight function and advection term and its applications

This paper is absorbed in the eigenvalue problem with indefinite weight function and advection term. We establish some results (one can see Theorem 1.2 for more details) to answer completely the questions concerning the positive eigenfunctions on the following eigenvalue problem with the flux boundary conditions: <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column minus d phi Subscript x x Baseline plus a phi Subscript x Baseline equals lamda m left-parenthesis x right-parenthesis phi comma 2nd Column a m p semicolon 0 greater-than x greater-than upper L comma 2nd Row 1st Column minus d phi Subscript x Baseline left-parenthesis 0 right-parenthesis plus a phi left-parenthesis 0 right-parenthesis equals minus b Subscript u Baseline a phi left-parenthesis 0 right-parenthesis comma 2nd Column a m p semicolon d phi Subscript x Baseline left-parenthesis upper L right-parenthesis minus a phi left-parenthesis upper L right-parenthesis equals minus b Subscript d Baseline a phi left-parenthesis upper L right-parenthesis period EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mo> − </mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>x</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi> λ </mml:mi> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi> ϕ </mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace"/> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mn>0</mml:mn> <mml:mo>&gt;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>L</mml:mi> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mo> − </mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>a</mml:mi> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>u</mml:mi> </mml:msub> <mml:mi>a</mml:mi> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mi>d</mml:mi> <mml:msub> <mml:mi> ϕ </mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> − </mml:mo> <mml:mi>a</mml:mi> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mi>a</mml:mi> <mml:mi> ϕ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>L</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \begin {cases} -d\phi _{xx}+a\phi _x=\lambda m(x)\phi ,\,&amp;0&gt;x&gt;L,\\ -d\phi _x(0)+a\phi (0)=-b_ua\phi (0),\;&amp;d\phi _x(L)-a\phi (L)=-b_da\phi (L). \end{cases} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> Furthermore, we give two examples to show the applications of Theorem 1.2, i.e., by employing eigenvalue theory, the theory of monotone dynamical system and the method of upper-lower solution, we obtained the global dynamic behavior of a single population model (3.4) and two-species competitive model (3.6) when the boundary condition parameters vary in all different situations. It is worth noting that it is not trivial to derive these conclusions in this paper.