Solutions of the autonomous Kirchhoff type equations in RN |
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Institution: | 1. School of Mathematics and Computer Science, Shanxi Datong University, Datong 037009, Shanxi, PR China;2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, PR China;3. School of Science, Tianjin University of Commerce, Tianjin 300134, PR China |
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Abstract: | Recently, the existence, the multiplicity and the concentration of solutions for the Kirchhoff type equations or systems have been extensively established by a number of authors. Such problems contain a nonlocal term which implies that the equation or system is no longer a pointwise identity. Thus, some researchers think that the nonlocal phenomenon will cause some mathematical difficulties. In this note, however, we will give a simple transformation so that the solutions of the autonomous Kirchhoff type equation or system are easily obtained by using the known solutions of the corresponding local equation or system. In particular, some qualitative properties of solutions for the local problems are also inherited. |
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Keywords: | Nonlocal problems Kirchhoff type equation Sobolev embedding theorem Qualitative property Variational method |
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