... Mλa,bΩ and uλ/∈M0λa,bΩ.ThenJλa,buλ0 in H−1Ω.Proof. Our proof is almost the same as that in Brown and Zhang 9,Theorem2.3or seeBinding et al. 31.Lemma 2.3. We have the ... J0 and α∞0Ω,respectively. The existence of ground state solutions of Eb is affected by the shape of the domainΩ and bx that satisfies some suitable conditions and has been the focus of ... the proof of Theorem 1.1.ByTheorem3.4,weobtainthatEλa,b has apositive solution uλ in H10Ω.4. Proof of Theorem 1.2 In this section, we will establish the existence of the second positive...