Sobolev’s inequality for double phase functionals with variable exponents

Abstract

AbstractOur aim in this paper is to establish generalizations of Sobolev’s inequality for double phase functionals {\Phi(x,t)=t^{p(x)}+a(x)t^{q(x)}}, where {p(\,{\cdot}\,)} and {q(\,{\cdot}\,)} satisfy log-Hölder conditions and {a(\,{\cdot}\,)} is nonnegative, bounded and Hölder continuous of order {\theta\in(0,1]}.