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)φp(u′)+b(t)φq(u′))′=λh(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 , φm(s)≔∣s∣m−2s{\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∈L1((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)≥0f\left(0)\ge 0) and semipositone (i.e., f(0)<0f\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.

Keywords

• double phase problem
• positive solution
• existence
• multiplicity
• 34b18
• 34l30
• 35j60