Existence of critical points for noncoercive functionals with critical Sobolev exponent. Issue 15 (13th October 2022) - Li, Zhouxin; Yuan, Xiang; Zhang, Qi
Abstract : We consider the following noncoercive functional with the critical Sobolev exponent 1 I ( u ) = 1 2 ∫ Ω 1 ( 1 + | u | ) 2 α | ∇ u | 2 d x − λ q ∫ Ω | u | q d x − 1 2 ∗ ( 1 − α ) ∫ Ω | u | 2 ∗ ( 1 − α ) d x, where λ > 0, 0 < α < ( N + 2 ) / 2 N, 2 < q < 2 ∗ ( 1 − α ), 2 ∗ = 2 N / ( N − 2 ) . We prove the existence of a nontrivial critical point via the mountain pass theorem and the multiplicity via Clark's theorem.