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Nodal solutions for a supercritical semilinear problem with variable exponent

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Abstract

In this paper, we are concerned with the following nonlinear supercritical elliptic problem with variable exponent,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u=|u|^{2^*+|x|^\alpha -2}u,~&{}\text {in}~ B_1(0),\\ u=0,\quad &{}\text {on} ~\partial B_1(0), \end{array}\right. } \end{aligned}$$

where \(2^*=\frac{2N}{N-2}\), \(0<\alpha <\min \{\frac{N}{2},N-2\}\), and \(B_1(0)\) is the unit ball in \(\mathbb {R}^N\), \(N\ge 3\). For any \(k\in \mathbb {N}\), we find, by variational methods, a pair of nodal solutions for this problem, which has exactly k nodal points.

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Acknowledgements

The authors would like to thank the referee for all insightful comments and valuable suggestions. Part of the work was done when the third author was visiting Academy of Mathematics and Systems Science, CAS. He would like to thank its nice hospitality. D. Cao was partially supported by NSFC Grants (Nos. 11771469 and 11688101). D. Cao was also supported by the Key Laboratory of Random Complex Strutures and Data Science. Z. Liu was supported by NSFC Grants (No. 11501166).

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510405, People’s Republic of China

    Daomin Cao

  2. Academy of Mathematics and Systems Science, CAS, Beijing, 100190, People’s Republic of China

    Daomin Cao & Shuanglong Li

  3. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, Henan, People’s Republic of China

    Zhongyuan Liu

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  1. Daomin Cao
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  2. Shuanglong Li
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  3. Zhongyuan Liu
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Correspondence to Zhongyuan Liu.

Additional information

Communicated by A. Malchiodi.

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Cao, D., Li, S. & Liu, Z. Nodal solutions for a supercritical semilinear problem with variable exponent. Calc. Var. 57, 38 (2018). https://doi.org/10.1007/s00526-018-1305-2

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