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Existence of positive solutions to a Laplace equation with nonlinear boundary condition

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Abstract

The positive solutions of a Laplace equation with a superlinear nonlinear boundary condition on a bounded domain are studied. For higher-dimensional domains, it is shown that non-constant positive solutions bifurcate from a branch of trivial solutions at a sequence of bifurcation points, and under additional conditions on nonlinearity, the existence of a non-constant positive solution for any sufficiently large parameter value is proved by using variational approach. It is also proved that for one-dimensional domain, there is only one bifurcation point, all non-constant positive solutions lie on the bifurcating curve, and for large parameter values, there exist at least two non-constant positive solutions. For a special case, there are exactly two non-constant positive solutions.

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Author information

Authors and Affiliations

  1. Department of Mathematics Education, Pusan National University, Busan, 609-735, Republic of Korea

    C.-G. Kim

  2. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Z.-P. Liang

  3. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    J.-P. Shi

Authors
  1. C.-G. Kim
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  2. Z.-P. Liang
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  3. J.-P. Shi
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Corresponding author

Correspondence to J.-P. Shi.

Additional information

C.-G. Kim is Partially supported by National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology, NRF-2011-357-C00006). Z.-P. Liang is Partially supported by National Natural Science Foundation of China (11571209, 11301313), Science Council of Shanxi Province (2015021007, 2013021001-4) and Shanxi 100 Talent program. J.-P. Shi is Partially supported by NSF grant DMS-1313243 and Shanxi 100 Talent program.

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Kim, CG., Liang, ZP. & Shi, JP. Existence of positive solutions to a Laplace equation with nonlinear boundary condition. Z. Angew. Math. Phys. 66, 3061–3083 (2015). https://doi.org/10.1007/s00033-015-0578-y

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  • DOI: https://doi.org/10.1007/s00033-015-0578-y

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