Fractional Calculus and Applied Analysis· 2026Q1
Anisotropic variable exponent Sobolev spaces and applications to elliptic equations with critical growth
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- 2026year
Short summary
New critical embeddings for anisotropic Sobolev spaces with variable exponents are established, resolving an open problem and enabling proofs of nontrivial weak solutions for anisotropic Leray-Lions equations with critical growth.
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Key points
- Established critical embeddings for anisotropic Sobolev spaces with variable exponents, resolving an open problem.
- Derived a Poincaré-type inequality at the harmonic-mean critical scale.
- Proved compactness criteria and a concentration–compactness principle for critical exponent attainment.
- Applied results to anisotropic Leray-Lions equations, proving existence of nontrivial weak solutions for large parameters.
- Demonstrated existence of at least m distinct solution pairs under oddness assumptions and sufficient parameter thresholds.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract We establish critical embeddings for anisotropic Sobolev spaces with variable exponents, resolving, under log-Hölder continuity supplemented by finite Sobolev regularity and a quantitative integrability-gap condition, an open problem raised by Fan. We derive a Poincaré-type inequality at the harmonic-mean critical scale, give compactness criteria when the critical exponent is attained on a nontrivial set, and prove a corresponding concentration–compactness principle. These results are then applied to anisotropic Leray–Lions equations with critical growth. Under a global quantitative superlinearity condition, the problem possesses a nontrivial weak solution for a sufficiently large reaction parameter. Under the additional oddness assumptions, for every prescribed integer m , it possesses at least m distinct pairs of nontrivial weak solutions once the parameter exceeds a sufficient threshold depending on m .
The authors' abstract, as published at the source. Fractional Calculus and Applied Analysis, 2026 · DOI ↗
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Field: Applied Mathematics
Applied MathematicsMathematics