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arXiv · 2106.04458

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

Prashanta Garain·Alexander Ukhlov
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Abstract

In this article, we consider mixed local and nonlocal Sobolev $(q,p)$-inequalities with extremal in the case $0<q<1<p<\infty$. 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 $p$-Laplace 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.

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Prashanta Garain, Alexander Ukhlov. 2021-06-08. Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems. https://arxiv.org/abs/2106.04458

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