Positive solutions for a $2n$th-order $p$-Laplacian boundary value problem involving all even derivatives
Keywords
p-Laplacian equation, Jensen's inequality, positive solution, fixed point index, coneAbstract
In this paper, we investigate the existence and multiplicity of positive solutions for the following $2n$th-order $p$-Laplacian boundary value problem $$ \cases -(((-1)^{n-1}x^{(2n-1)})^{p-1})'\\ =f(t,x,-x^{\prime\prime},\ldots,(-1)^{n-1}x^{(2n-2)}) &\text{for } t\in [0,1], \\ x^{(2i)}(0)=x^{(2i+1)}(1)=0 & \text{for } i=0,\ldots,n-1, \endcases $$ where $n\ge 1$ and $f\in C([0,1]\times \mathbb{R}_+^{n}, \mathbb{R}_+)(\mathbb{R}_+:=[0,\infty))$ depends on $x$ and all derivatives of even orders. Based on a priori estimates achieved by utilizing properties of concave functions and Jensen's integral inequalities, we use fixed point index theory to establish our main results. Moreover, our nonlinearity $f$ is allowed to grow superlinearly and sublinearly.Downloads
Published
2012-04-23
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1.
XU, Jiafa, WEI, Zhongli and DING, Youzheng. Positive solutions for a $2n$th-order $p$-Laplacian boundary value problem involving all even derivatives. Topological Methods in Nonlinear Analysis. Online. 23 April 2012. Vol. 39, no. 1, pp. 23 - 36. [Accessed 14 December 2024].
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