TITLE

# Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential

AUTHOR(S)
Daomin Cao; Shusen Yan
PUB. DATE
July 2010
SOURCE
Calculus of Variations & Partial Differential Equations;Jul2010, Vol. 38 Issue 3/4, p471
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: under the assumptions that N â‰¥ 7, $${\mu\in \left[0,\frac{(N-2)^2}4-4\right)}$$ and a > 0, where $${2^{\ast}=\frac{2N}{N-2}}$$ , and Î© is an open bounded domain in $${\mathbb{R}^N}$$ which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth,where $${\varepsilon_{n} > 0}$$ is small and $${\varepsilon_n \to 0}$$ as n â†’ + âˆž. By the critical point theory for the even functionals, for each fixed $${\varepsilon_{n} > 0}$$ small, (**) n has a sequence of solutions $${u_{k,\varepsilon_{n}} \in H^{1}_{0}(\Omega)}$$ . We obtain the existence of infinitely many solutions for (*) by showing that as n â†’ âˆž, $${u_{k,\varepsilon_{n}}}$$ converges strongly in $${H^{1}_{0}(\Omega)}$$ to u k, which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of $${\{u_{k,\varepsilon_n}\}}$$.
ACCESSION #
50354931

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