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Constant-sign and sign-changing solutions for the Sturm–Liouville boundary value problems

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Abstract

In this paper, we study the Strum–Liouville boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} -(p(x)u^{\prime }(x))^{\prime }+q(x)u(x)=f(x,u(x)), &{}\quad 0\le x\le 1,\\ \alpha u^{\prime }(0)-\beta u(0)=0, &{}\quad \gamma u^{\prime }(1)+\sigma u(1)=0.\end{array} \right. \end{aligned}$$

Using critical point theory and suitable truncation techniques, we prove the existence of two opposite constant sign solutions and infinitely many sign-changing solutions of the problem. Different from the existing research, we do not impose any restrictions to the behavior of the nonlinear term \(f\) at infinity.

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Acknowledgments

The authors thank the referee for his careful reading of the paper and his valuable comments.

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Authors and Affiliations

  1. College of Computation Science, Zhongkai University of Agriculture and Engineering, Guangzhou, 510225, Guangdong, People’s Republic of China

    Tieshan He, Zhaohong Sun & Hongming Yan

  2. College of Chemistry and Chemical Engineering, Zhongkai University of Agriculture and Engineering, Guangzhou, 510225, Guangdong, People’s Republic of China

    Yimin Lu

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  1. Tieshan He
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  2. Zhaohong Sun
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  3. Hongming Yan
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  4. Yimin Lu
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Correspondence to Yimin Lu.

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Communicated by A. Constantin.

Supported by the NNSF of China (No. 11201504) and the Scientific Research Plan Item of Guangdong Provincial Department of Education (No. 2011TJK468).

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He, T., Sun, Z., Yan, H. et al. Constant-sign and sign-changing solutions for the Sturm–Liouville boundary value problems. Monatsh Math 179, 41–55 (2016). https://doi.org/10.1007/s00605-014-0694-3

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