Analysis of positive solutions to one-dimensional generalized double phase problems

Abstract 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.