Interior and boundary regularity results for strongly nonhomogeneousp,q-fractional problems

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:math> {(p,q)} -Laplacian, denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msubsup><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mo>-</m:mo><m:mi mathvariant="normal">Δ</m:mi></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mi>p</m:mi><m:msub><m:mi>s</m:mi><m:mn>1</m:mn></m:msub></m:msubsup><m:mo>+</m:mo><m:msubsup><m:mrow><m:mo stretchy="false">(</m:mo><m:mrow><m:mo>-</m:mo><m:mi mathvariant="normal">Δ</m:mi></m:mrow><m:mo stretchy="false">)</m:mo></m:mrow><m:mi>q</m:mi><m:msub><m:mi>s</m:mi><m:mn>2</m:mn></m:msub></m:msubsup></m:mrow></m:math> {(-Δ)^{s₁}ₚ+(-Δ)^{s₂}q} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:msub><m:mi>s</m:mi><m:mn>2</m:mn></m:msub><m:mo>,</m:mo><m:msub><m:mi>s</m:mi><m:mn>1</m:mn></m:msub></m:mrow><m:mo>∈</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mn>0</m:mn><m:mo>,</m:mo><m:mn>1</m:mn><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {s₂,s₁∈(0,1)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mn>1</m:mn><m:mo><</m:mo><m:mi>p</m:mi></m:mrow><m:mo>,</m:mo><m:mrow><m:mi>q</m:mi><m:mo><</m:mo><m:mi mathvariant="normal">∞</m:mi></m:mrow></m:mrow></m:math> {1<p,q<∞} . We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>q</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:math> {(p,q)} -problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish a new Hopf-type maximum principle and a strong comparison principle in both situations.