The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

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Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth (2006, unknown): “integral with the boundary values 0 and a> 0. Then ua is a solution with the same boundary values by Theorem 5.7 of [ 11 ].”
Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent (2008, background): “then the smooth functions are dense in W 1,p(·)(Ω) and so the Sobolev space with zero boundary values, denoted by W 1,p(·) 0 (Ω), as the closure of C ∞ 0 (Ω) under the norm ‖ · ‖, are meaningful, see [18,20].”
Global Nonexistence for Nonlinear Kirchhoff Systems (2010, background): “Details, extensions and further references can be found in [10,12–16].”
WEAK SOLUTIONS FOR ANISOTROPIC NONLINEAR ELLIPTIC EQUATIONS WITH VARIABLE EXPONENTS (2009, unknown): “) 0 (Ω), where C depends on p, |Ω|, diam(Ω) and N (see [12]), and so ‖u‖ := |∇u|p(.”
Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent (2011, unknown): “For more details concerning the variable exponent spaces W 1,pð Þ 0 ð Þ we refer to the papers [9,38–41].”