Acta mathematica scientia, Series B >
MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER
Received date: 2007-12-29
Online published: 2010-01-20
Supported by
Project supported by the National Science Foundation of China (10471047) and the Natural Science Foundation of Guangdong Province (04020077).
Some embedding inequalities in Hardy-Sobolev space are proved. Furthermore, by the improved inequalities and the linking theorem, in a new k-order Sobolev-Hardy space, we obtain the existence of sign-changing solutions for the nonlinear elliptic equation
\left\{\begin{array}{ll}
\disp -\Delta_{(k)}u:=-\Delta
u-\frac{(N-2)^2}{4}\frac{u}{|x|^2}-\frac{1}{4}\sum\limits_{i=1}^{k-1}\frac{u}{|x|^2(\ln_{(i)}R/|x|)^2}
=f(x,u),&\quad x\in\Omega,
u=0,&\quad x\in\partial\Omega, \end{array} \right.
where 0\in \Omega \subset B_a(0)\subset {\Bbb R}^N, N\geq 3, \ln_{(i)}=\prod\limits_{j=1}^i\ln^{(j)}, and R=ae^{(k-1)}, where e^{(0)}=1, e^{(j)}=e^{e^{(j-1)}} for j\geq 1, \ln^{(1)}=\ln, \ln^{(j)}=\ln\ln^{(j-1)} for j\geq 2. Besides, positive and negative solutions are obtained by a variant mountain pass theorem.
Key words： nonlinear elliptic equation; critical potential; linking
WANG You-Jun , SHEN Yao-Tian . MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER[J]. Acta mathematica scientia, Series B, 2010 , 30(1) : 113 -124 . DOI: 10.1016/S0252-9602(10)60027-6
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