Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

By P. Garain, A. Ukhlov. Nonlinear Analysis, 2021

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Abstract. In this article, we consider mixed local and nonlocal Sobolev (q, p)-inequalities with extremal in the case 0 < q < 1 < p < ∞. We prove that the extremal of such inequalities is unique up to a multiplicative constant that is associated with a singular elliptic problem involving the mixed local and nonlocal p-Laplace operator. Moreover, it is proved that the mixed Sobolev inequalities are necessary and sufficient condition for the existence of weak solutions of such singular problems. As a consequence, a relation between the singular pLaplace and mixed local and nonlocal p-Laplace equation is established. Finally, we investigate the existence, uniqueness, regularity and symmetry properties of weak solutions for such problems.

Subfields: Applied Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/8560179df5faee7539e835f55453bcd75e4a5c4d

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In Applied Mathematics

In Mechanics of Materials

Citations in the corpus

Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations (2022, background): “In the case of a bounded function f , Biagi et al. [5] has obtained local Hölder continuity for globally bounded solutions and Garain-Ukhlov [31] studied existence, uniqueness, local boundedness and further qualitative properties of solutions.”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, unknown): “For this, we refer to Buccheri-da Silva-Miranda [12], da SilvaSalort [17], Biagi-Mugnai-Vecchi [8], Garain-Kinnunen [28] and Garain-Ukhlov [30].”