Abstract
We will be concerned with the existence result of unilateral problem associated to the equations of the form Au + g(x, u, ∇u) = f, where A is a Leray‐Lions operator from its domain into . On the nonlinear lower order term g(x, u, ∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, and satisfies the sign condition. The right‐hand side f belongs to .
Highlights
Let Ω be an open bounded subset of RN, N ≥ 2, with segment property
We will be concerned with the existence result of unilateral problem associated to the equations of the form Au + g(x,u,∇u) = f, where A is a Leray-Lions operator from its domain D(A) ⊂ W01LM(Ω) into W−1EM(Ω)
Au = − div a(x,u, ∇u) is a Leray-Lions operator defined on its domain Ᏸ(A) ⊂ W01LM(Ω), with M an N-function and where g is a nonlinearity with the “natural” growth condition g(x, s, ξ) ≤ b |s| h(x) + M |ξ|
Summary
Let Ω be an open bounded subset of RN , N ≥ 2, with segment property. Let us consider the following nonlinear Dirichlet problem:. Au = − div a(x,u, ∇u) is a Leray-Lions operator defined on its domain Ᏸ(A) ⊂ W01LM(Ω), with M an N-function and where g is a nonlinearity with the “natural” growth condition g(x, s, ξ) ≤ b |s| h(x) + M |ξ|. Which satisfies the classical sign condition g(x, s, ξ) · s ≥ 0. The right-hand side f belongs to W−1EM(Ω).
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