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Nodal solutions for Laplace equations with critical Sobolev and Hardy exponents on $R^N$

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  • This paper is concerned with the existence and nodal character of the nontrivial solutions for the following equations involving critical Sobolev and Hardy exponents:

    $-\Delta u + u - \mu \frac{u}{|x|^2}=|u|^{2^*-2}u + f(u),$
    $u \in H^1_r (\R ^N),(1)$

    where $2^$*$=\frac{2N}{N-2}$ is the critical Sobolev exponent for the embedding $H^1_r (\R ^N) \rightarrow L^{2^}$*$ (\R ^N)$, $\mu \in [0, \ (\frac {N-2}{2})^2)$ and $f: \R \rightarrow\R $ is a function satisfying some conditions. The main results obtained in this paper are that there exists a nontrivial solution of equation (1) provided $N\ge 4$ and $\mu \in [0, \ (\frac {N-2}{2})^2-1] $ and there exists at least a pair of nontrivial solutions $u^+_k$, $u^-_k$ of problem (1) for each k $\in N \cup \{0\}$ such that both $u^+_k$ and $u^-_k$ possess exactly k nodes provided $N\ge 6$ and $\mu \in [0, \ (\frac {N-2}{2})^2-4]$.

    Mathematics Subject Classification: 35J25, 35J60, 35J65.


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