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Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity

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Abstract

Let B ⊂ ℝn be the unit ball centered at the origin. The authors consider the following biharmonic equation:

$$\left\{ {\begin{array}{*{20}{c}} {{\Delta ^2}u = \lambda {{\left( {1 + u} \right)}^p}}&{in \mathbb{B},} \\ {u = \frac{{\partial u}}{{\partial \nu }} = 0}&{on\partial \mathbb{B},} \end{array}} \right.$$

where \(p > \frac{{n + 4}}{{n - 4}}\) and v is the outward unit normal vector. It is well-known that there exists a λ* > 0 such that the biharmonic equation has a solution for λ ∈ (0, λ*) and has a unique weak solution u* with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies \(u* \leqslant {r^{ - \frac{4}{{p - 1}}}} - 1\) on the unit ball, which actually solve a part of the open problem left in [Dàvila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193].

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Acknowledgement

The first author would like to thank his advisor Prof. Yi Li for his constant support and encouragement.

Author information

Correspondence to Baishun Lai.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11201119, 11471099), the International Cultivation of Henan Advanced Talents and the Research Foundation of Henan University (No. yqpy20140043).

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Lai, B., Yan, Z. & Zhang, Y. Singularity of the extremal solution for supercritical biharmonic equations with power-type nonlinearity. Chin. Ann. Math. Ser. B 38, 815–826 (2017). https://doi.org/10.1007/s11401-017-1097-2

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Keywords

  • Minimal solutions
  • Regularity
  • Stability
  • Fourth order

2000 MR Subject Classification

  • 35B45
  • 35J40