Blowing-up solutions concentrated along minimal submanifolds for some supercritical Hamiltonian systems on Riemannian manifolds

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left({\mathcal{ {\mathcal M} }},g) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">K</m:mi> <m:mo>,</m:mo> <m:mi>κ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left({\mathcal{K}},\kappa ) be two Riemannian manifolds of dimensions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>N</m:mi> </m:math> N and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> </m:math> m , respectively. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \omega \in {C}^{2}\left({\mathcal{ {\mathcal M} }}) satisfy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:math> \omega \gt 0 . The warped product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> <m:msub> <m:mrow> <m:mo>×</m:mo> </m:mrow> <m:mrow> <m:mi>ω</m:mi> </m:mrow> </m:msub> <m:mi class="MJX-tex-caligraphic" mathvariant="script">K</m:mi> </m:math> {\mathcal{ {\mathcal M} }}{\times }_{\omega }{\mathcal{K}} is the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>N</m:mi> <m:mo>+</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(N+m) -dimensional product manifold <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> <m:mo>×</m:mo> <m:mi class="MJX-tex-caligraphic" mathvariant="script">K</m:mi> </m:math> {\mathcal{ {\mathcal M} }}\times {\mathcal{K}} equipped with the metric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>κ</m:mi> </m:math> g+{\omega }^{2}\kappa . We consider the following elliptic system: (1) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mfenced open="{" close=""> <m:mrow> <m:mtable displaystyle="true"> <m:mtr> <m:mtd columnalign="left"> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>Δ</m:mi> </m:mrow> <m:mrow> <m:mi>g</m:mi> <m:mo>+</m:mo> <m:msup> <m:mrow> <m