MULTIPLE AND SIGN-CHANGING SOLUTIONS FOR NONLINEAR ELLIPTIC EQUATION WITH CRITICAL POTENTIAL AND CRITICAL PARAMETER

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doi:10.1016/S0252-9602(10)60027-6

Abstract:

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 - Δ_{(k)} u := - Δ u - \frac{(N - 2)^{2}}{4} \frac{u}{| x |^{2}} - \frac{1}{4} \sum_{i = 1}^{k - 1} \frac{u}{| x |^{2} (\ln_{(i)} R / | x |)^{2}} = f (x, u), & x \in Ω, u = 0, & x \in \partial Ω, \end{array}

$\begin{array}{ll} \disp -\Delta_{(k)}u:=-\Delta u-\frac{(N-2)^2}{4}\frac{u}{|x|^2}-\frac{1}{4}\sum\limits_{i=1}^{k-1}\frac{u}{|x|^2(\ln_{(i)}R/|x|)^2} =f(x,u),&\quad x\in\Omega, u=0,&\quad x\in\partial\Omega, \end{array}$ \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

CLC Number:

46E35

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.

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