ABSTRACT

Let Ω be a domain in Թ௡ and ݈݁ݐ ݌ be a real number that satisfies 1൏݌ .∞൏We denote by ൫ܹ଴ ଵ,௣ሺΩሻ, ‖ ‖଴,௣,׏൯ the classical Sobolev space. The ݌ െLaplacian on ܹ଴ ଵ,௣ሺΩሻ is the operator െΔ௣: ܹ଴ ଵ,௣ሺΩሻ → ቀܹ଴ ଵ,௣ሺΩሻቁ ∗ defined as follows: ݑ଴ܹ∋ ଵ,௣ሺΩሻ → െΔ௣ݑ ∋ ቀܹ଴ ଵ,௣ሺΩሻቁ ∗ where |ݑ׏| ׬ ≕ 〈ݒ ,ݑ௣െΔ〈 ଴ܹ∋ݒ each for ݔ݀ ஐ ݒ׏ ∙ ݑ׏ଶ௣ି ଵ,௣ሺΩሻ. Existence results for two types of equations, namely െΔ௣ݑ ൌ ܨ ∋ ቀܹ଴ ଵ,௣ሺΩሻቁ ∗ and െΔ௣ݑൌ݂ሺݔ ,ݑሻ will be presented. Here, ݂: Ω ൈ Թ → Թ is a Carathéodory function satisfying a growth condition, so that ݂ሺݔ ,ݑሻ can be identified with an element in ቀܹ଴ ଵ,௣ሺΩሻቁ ∗ for each ݑ଴ܹ∋ ଵ,௣ሺΩሻ. These existence results are obtained by three types of methods: 1. The operator െΔ௣ is regarded as a duality mapping and then abstract surjectivity results for duality mappings are used. 2. The second method relies on the basic properties of the Leray-Schauder topological degree. 3. A direct variational method and the “Mountain Pass Theorem” of Ambrosetti and Rabinowitz can be also used.

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