Multiplicity and uniqueness of positive solutions for nonhomogeneous semilinear elliptic equation with critical exponent

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• 作者：Na Ba ; Yan-yan Wang ; Lie Zheng
• 关键词：nonhomogeneous semilinear elliptic problems ; multiplicity ; uniqueness
• 刊名：Acta Mathematicae Applicatae Sinica, English Series
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：32
• 期：1
• 页码：81-94
• 全文大小：275 KB
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• 作者单位：Na Ba (1)
  Yan-yan Wang (2)
  Lie Zheng (3)
  
  1. School of Science, Hubei University of Technology, Wuhan, 430068, China
  2. Department of Mathematics, Zhoukou Normal University, Zhoukou, 466001, China
  3. School of Science, Hubei University of Technology, Wuhan, 430068, China
  
• 刊物主题：Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics;
• 出版者：Institute of Applied Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
• ISSN：1618-3932

文摘

In this paper, we consider the following problem $$ \left\{ {\begin{array}{*{20}{c}}{ - \Delta u\left( x \right) + u\left( x \right) = \lambda \left( {{u^p}\left( x \right) + h\left( x \right)} \right),\;x \in {\mathbb{R}^N},} \\ {u\left( x \right) \in {H^1}\left( {{\mathbb{R}^N}} \right),\;u\left( x \right) \succ 0,\;x \in {\mathbb{R}^N},\;} \end{array}} \right.\;\left( * \right)$$, where λ > 0 is a parameter, p = (N+2)/(N−2). We will prove that there exists a positive constant 0 < λ* < +∞ such that (*) has a minimal positive solution for λ ∈ (0, λ*), no solution for λ > λ*, a unique solution for λ = λ*. Furthermore, (*) possesses at least two positive solutions when λ ∈ (0, λ*) and 3 ≤ N ≤ 5. For N ≥ 6, under some monotonicity conditions of h we show that there exists a constant 0 < λ** < λ* such that problem (*) possesses a unique solution for λ ∈ (0, λ**). Keywords nonhomogeneous semilinear elliptic problems multiplicity uniqueness 2000 MR Subject Classification 35J20 35J60 Na Ba was supported by the National Natural Science Foundation of China(No. 11201132) and Scientific Research Foundation for Ph.D of Hubei University of Technology (No. BSQD12065). Lie Zheng was supported by the Science Research Project of Hubei Provincial Department of education (No. d200614001).