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

Abstract

The aim of this paper is to prove that (1.1) has infinitely many solutions if N ≥ 7 and 0 ≤ μ < (N-2)²/4 − 4. As in [13] one of the major difficulty to prove the existence of infinitely many solutions for (1.1) by using the variational methods is that I (u) does not satisfy the Palais–Smale condition for large energy level, since 2* is the critical exponent for the Sobolev embedding from H¹(Ω) to Lq(Ω). Another difficulty is that, unlike in [13], every nontrivial solution of (1.1) is singular at x = 0 if μ ≠ 0 (see [8,9]). So, different techniques are needed to deal with the case μ > 0.