On Elliptic Equations in Orlicz Spaces Involving Natural Growth Term and Measure Data

and Applied Analysis 3 2. Preliminaries LetM : R → R be anN-function, that is,M is continuous, convex, withM t > 0 for t > 0, M t /t → 0 as t → 0 and M t /t → ∞ as t → ∞. The N-function M conjugate to M is defined byM t sup{st −M s : s > 0}. Let P and Q be two N-functions. P Q means that P grows essentially less rapidly than Q; that is, for each ε > 0, P t Q εt −→ 0 as t −→ ∞. 2.1 The N-function M is said to satisfy the Δ2 condition if for some k > 0: M 2t ≤ kM t for all t ≥ 0; when this inequality holds only for t ≥ t0 > 0,M is said to satisfy the Δ2 condition near infinity. LetΩ be an open subset ofR . The Orlicz classLM Ω resp., the Orlicz space LM Ω is defined as the set of equivalence classes of real-valued measurable functions u onΩ such that ∫ Ω M u x dx < ∞ resp. ∫ Ω M u x /λ dx < ∞ for some λ > 0 . Note that LM Ω is a Banach space under the norm ‖u‖M,Ω inf{λ > 0 : ∫ Ω M u x /λ dx ≤ 1} and LM Ω is a convex subset of LM Ω . The closure in LM Ω of the set of bounded measurable functions with compact support in Ω is denoted by EM Ω . In general EM Ω / LM Ω and the dual of EM Ω can be identified with LM Ω by means of the pairing ∫ Ω u x v x dx, and the dual norm on LM Ω is equivalent to ‖ · ‖M,Ω. We now turn to the Orlicz-Sobolev space. WLM Ω resp. WEM Ω is the space of all functions u such that u and its distributional derivatives up to order 1 lie in LM Ω resp. EM Ω . This is a Banach space under the norm ‖u‖1,M,Ω ∑ |α|≤1 ‖Du‖M,Ω. Thus WLM Ω and WEM Ω can be identified with subspaces of the product of N 1 copies of LM Ω . Denoting this product by ΠLM, we will use the weak topologies σ ΠLM,ΠEM and σ ΠLM,ΠLM . The space W 1 0EM Ω is defined as the norm closure of the Schwartz space D Ω in WEM Ω and the space W1 0LM Ω as the σ ΠLM,ΠEM closure of D Ω in WLM Ω . We say that un converges to u for the modular convergence in WLM Ω if for some λ > 0, ∫ Ω M D un − Du /λ dx → 0 for all |α| ≤ 1. This implies convergence for σ ΠLM,ΠLM . If M satisfies the Δ2 condition on R near infinity only when Ω has finite measure , then modular convergence coincides with norm convergence. Fore more details about the Orlicz spaces and their properties one can see 11, 12 . For k > 0, we define the truncation at height k, Tk : R → R by: Tk s max −k, min k, s . 3. Main Result 3.1. Useful Results First, we give the following definitions and results which will be used in our main result. The p-capacity Cp B,Ω of any set B ⊂ Ω with respect to Ω is defined in the following classical way. The p-capacity of any compact set K ⊂ Ω is first defined as Cp K,Ω inf {∫ Ω ∣∇φ∣pdx : φ ∈ D Ω , φ ≥ χK } , 3.1 4 Abstract and Applied Analysis where χK is the characteristic function of K; we will use the convention that infφ ∞. The p-capacity of any open subset U ⊂ Ω is then defined by Cp U,Ω sup { Cp K,Ω , K compact K ⊂ Ω } . 3.2 Finally, the p-capacity of any subset B ⊂ Ω is defined by Cp B,Ω inf { Cp U,Ω , U open B ⊂ U } . 3.3 Definition 3.1. We say that u is a weak solution of the problem ( Pμ ) if u is measurable, Tk u ∈ W1 0LM Ω , ∫