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

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: $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2}u+a u\quad {\rm in}\;\Omega,\quad u=0 \quad {\rm on}\; \partial\Omega,\qquad (*)$$ under the assumptions that N ≥ 7, \({\mu\in \left[0,\frac{(N-2)^2}4-4\right)}\) and a > 0, where \({2^{\ast}=\frac{2N}{N-2}}\) , and Ω 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, $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2-\varepsilon_n}u+au \quad {\rm in}\,\Omega, \quad u=0 \quad {\rm on}\;\partial\Omega,\qquad(\ast\ast)_n$$ where \({\varepsilon_{n} > 0}\) is small and \({\varepsilon_n \to 0}\) as n → + ∞. 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 → ∞, \({u_{k,\varepsilon_{n}}}\) converges strongly in \({H^{1}_{0}(\Omega)}\) to uk, 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 \({\{u_{k,\varepsilon_n}\}}\).