Positive solutions for a 2nth-order p-Laplacian boundary value problem involving all derivatives

Electronic Journal of Differential Equations

关键词: Integro-ordinary differential equation;    a priori estimate;    index;    fixed point;    positive solution;   

DOI  :  

来源: DOAJ

【 摘 要 】

In this work, we are mainly concerned with the positivesolutions for the 2nth-order p-Laplacian boundary-valueproblem$$displaylines{-(((-1)^{n-1}x^{(2n-1)})^{p-1})'=f(t,x,x',ldots,(-1)^{n-1}x^{(2n-2)},(-1)^{n-1}x^{(2n-1)}),crx^{(2i)}(0)=x^{(2i+1)}(1)=0,quad (i=0,1,ldots,n-1),}$$where $nge 1$ and $fin C([0,1]imes mathbb{R}_+^{2n},mathbb{R}_+)(mathbb{R}_+:=[0,infty))$.To overcome the difficulty resulting from all derivatives,we first convert the above problem into a boundary value problemfor an associated second order integro-ordinary differential equationwith p-Laplacian operator. Then, by virtue of the classic fixedpoint index theory, combined with a priori estimates of positive solutions,we establish some results on the existence and multiplicity of positivesolutions for the above problem. Furthermore, our nonlinear term f isallowed to grow superlinearly and sublinearly.

【 授权许可】

Unknown