Abstract
We extend classical Troisi’s inequality to the weighted p-Sobolev spaces on stretched cone, edge, and corner respectively. The results here can be used to investigate anisotropic elliptic equations involving cone degeneracy, edge degeneracy, and corner degeneracy, which will be studied in our forthcoming papers.
Highlights
Its classical form can be described as: given 1 ≤ pi < ∞, i = 1, . . . , n, for a smooth function u compactly supported in Rn, the following inequality holds: n u s≤C
Applications of (1.1) can be found in [3] to study the existence of fundamental solutions to anisotropic elliptic equations. Another generalization of (1.1) in [4] is used to prove regularity of the weak solution to the Navier–Stokes equations based on one component of velocity
By arithmetic and geometric mean inequality, (1.1) becomes an anisotropic Sobolev inequality presented as u s≤
Summary
Troisi (cf [1]) found an important inequality. N, for a smooth function u compactly supported in Rn, the following inequality holds:. Troisi’s inequality that can be used to study the existence of multiple nonnegative solutions to the anisotropic critical problem (cf [2]). Is anisotropic critical exponent and max1≤i≤n{pi} < s. Applications of (1.1) can be found in [3] to study the existence of fundamental solutions to anisotropic elliptic equations. Another generalization of (1.1) in [4] is used to prove regularity of the weak solution to the Navier–Stokes equations based on one component of velocity. By arithmetic and geometric mean inequality, (1.1) becomes an anisotropic Sobolev inequality presented as u s≤
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