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Analysis of positive solutions to one-dimensional generalized double phase problems
Advances in Nonlinear Analysis ( IF 3.2 ) Pub Date : 2022-04-30 , DOI: 10.1515/anona-2022-0240
Byungjae Son 1 , Inbo Sim 2
Affiliation  

We study positive solutions to the one-dimensional generalized double phase problems of the form: ( a ( t ) φ p ( u ) + b ( t ) φ q ( u ) ) = λ h ( t ) f ( u ) , t ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , \left\{\begin{array}{l}-(a\left(t){\varphi }_{p}\left(u^{\prime} )+b\left(t){\varphi }_{q}\left(u^{\prime} ))^{\prime} =\lambda h\left(t)f\left(u),\hspace{1em}t\in \left(0,1),\\ u\left(0)=0=u\left(1),\end{array}\right. where 1 < p < q < 1\lt p\lt q\lt \infty , φ m ( s ) s m 2 s {\varphi }_{m}\left(s):= | s{| }^{m-2}s , a , b C ( [ 0 , 1 ] , [ 0 , ) ) a,b\in C\left(\left[0,1],{[}0,\infty )) , h L 1 ( ( 0 , 1 ) , ( 0 , ) ) C ( ( 0 , 1 ) , ( 0 , ) ) , h\in {L}^{1}\left(\left(0,1),\left(0,\infty ))\cap C\left(\left(0,1),\left(0,\infty )), and f C ( [ 0 , ) , R ) f\in C\left({[}0,\infty ),{\mathbb{R}}) is nondecreasing. More precisely, we show various existence results including the existence of at least two or three positive solutions according to the behaviors of f ( s ) f\left(s) near zero and infinity. Both positone (i.e., f ( 0 ) 0 f\left(0)\ge 0 ) and semipositone (i.e., f ( 0 ) < 0 f\left(0)\lt 0 ) problems are considered, and the results are obtained through the Krasnoselskii-type fixed point theorem. We also apply these results to show the existence of positive radial solutions for high-dimensional generalized double phase problems on the exterior of a ball.

中文翻译:

一维广义双相问题正解分析

我们研究以下形式的一维广义双相问题的正解: - ( 一种 ( ) φ p ( 或者 ' ) + b ( ) φ 什么 ( 或者 ' ) ) ' = λ H ( ) F ( 或者 ) , ( 0 , ) , 或者 ( 0 ) = 0 = 或者 ( ) , \left\{\begin{数组}{l}-(a\left(t){\varphi }_{p}\left(u^{\prime} )+b\left(t){\varphi }_ {q}\left(u^{\prime} ))^{\prime} =\lambda h\left(t)f\left(u),\hspace{1em}t\in \left(0,1) ,\\u\left(0)=0=u\left(1),\end{array}\right. 在哪里 < p < 什么 < 1\lt p\lt q\lt \infty , φ ( s ) s - s {\varphi }_{m}\left(s):= | s{| }^{m-2}s , 一种 , b C ( [ 0 , ] , [ 0 , ) ) a,b\in C\left(\left[0,1],{[}0,\infty )) , H 大号 ( ( 0 , ) , ( 0 , ) ) C ( ( 0 , ) , ( 0 , ) ) , h\in {L}^{1}\left(\left(0,1),\left(0,\infty ))\cap C\left(\left(0,1),\left(0,\无限)), F C ( [ 0 , ) , R ) f\in C\left({[}0,\infty ),{\mathbb{R}}) 是不减的。更准确地说,我们展示了各种存在结果,包括根据行为的至少两个或三个正解的存在。 F ( s ) f\left(s) 接近零和无穷大。两个正音(即, F ( 0 ) 0 f\左(0)\ge 0 )和半位置(即, F ( 0 ) < 0 f\左(0)\lt 0 ) 问题,通过 Krasnoselskii 型不动点定理得到结果。我们还应用这些结果来证明球外部的高维广义双相问题存在正径向解。
更新日期:2022-04-30
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