Proof. The proof is standard. Indeed, if γ(SCc) ≤ l, then, by virtue of Lemma 4.4, there would exist δ > 0 with δ < c such that γ(Isc+δ ∪ Dε; Is0 ∪ Dε, Is−1) ≤ γ(Isc−δ ∪ Dε; Is0 ∪ Dε, Is−1) + l. The definition of ck implies γ(Isc−δ∪Dε;Is0∪Dε,Is−1) ≤ k−1. Thus, we have γ(Isc+δ ∪ Dε; Is0 ∪ Dε, Is−1) ≤ k + l − 1, but this contradicts the definition of ck+l. Therefore, we obtain γ(SCc) ≥ l + 1. In a similar way, we can prove ck → +∞ as k → +∞. □ Proof of Theorem 1.3. Note that Lemma 4.8 in particular yields a sequence {±uk} of distinct solutions with Is(uk) → +∞ and uk ⊂ SCck ⊂ X0s(Ω) \ Dε. Hence, ‖uk‖ → +∞, and uk changes sign for every k, completing the proof of Theorem 1.3. □ Acknowledgements. This work is supported by NSFC grants No.11771170, 11701203, and 11328101, as well as a program for Changjiang Scholars and Innovative Research Team in University No. IRT13066 and the self-determined research funds of CCNU from the colleges? basic research and operation of MOE (No. CCNU16A05042).
