Abstract
We prove the existence results in the setting of Orlicz spaces for the following nonlinear elliptic equation: $$A(u)+g(x,u,Du)=\mu, $$ where A is a Leray-Lions operator defined on $D(A)\subset W_{0}^{1}L_{M}(\Omega)$ , while g is a nonlinear term having a growth condition with respect to Du, but does not satisfy any sign condition. The right-hand side μ is a bounded Radon measure data.
Highlights
Aharouch et al [ ] proved the existence results in the setting of Orlicz spaces for the unilateral problem associated to the following equation: A(u) + g(x, u, Du) = f, ( )
Dong and Fang Boundary Value Problems (2015) 2015:18 where A(u) = – div a(x, u, Du) is a Leray-Lions operator defined on D(A) ⊂ W LM( ), a and g satisfy the following growth conditions: (a ) |a(x, s, ξ )| ≤ c(x) + k P – (M(k |s|)) + k M – (M(k |ξ |)), k, k, k, k ≥, c(x) ∈ EM ( ), (g) |g(x, s, ξ )| ≤ γ (x) + ρ(s)M(|ξ |), γ (x) ∈ L ( ), and ρ : R → R+ is continuous, ρ ≥, ρ ∈ L (R), for almost every x ∈, for all s ∈ R, ξ ∈ RN, where M and P are N -functions such that P M
Where A(u) = – div a(x, u, Du) is a Leray-Lions operator defined on D(A) ⊂ W LM( ) having the following growth condition: a(x, s, ξ ) ≤ β c(x) + M – M |s| + M – M |ξ |, β >, c(x) ∈ EM ( )
Summary
Aharouch et al [ ] proved the existence results in the setting of Orlicz spaces for the unilateral problem associated to the following equation: A(u) + g(x, u, Du) = f , ( ) Dong and Fang Boundary Value Problems (2015) 2015:18 where A(u) = – div a(x, u, Du) is a Leray-Lions operator defined on D(A) ⊂ W LM( ), a and g satisfy the following growth conditions: (a ) |a(x, s, ξ )| ≤ c(x) + k P – (M(k |s|)) + k M – (M(k |ξ |)), k , k , k , k ≥ , c(x) ∈ EM ( ), (g) |g(x, s, ξ )| ≤ γ (x) + ρ(s)M(|ξ |), γ (x) ∈ L ( ), and ρ : R → R+ is continuous, ρ ≥ , ρ ∈ L (R), for almost every x ∈ , for all s ∈ R, ξ ∈ RN , where M and P are N -functions such that P M.
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