Liouville theorems of solutions to mixed order Hénon-Hardy type system with exponential nonlinearity

In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:math> R₊² : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:msup> <m:mrow> <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:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>a</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mi>v</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mspace width="1em" /> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <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>v</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>b</m:mi> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> <m:mi>v</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mspace width="0.17em" /> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mo>,</m:mo> <m:mspace width="1em" /> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> ${cases}{(-{Δ})}α /2u(x)= x{ }ᵃ{u}^{{p}₁}(x){e}^{{q}₁v(x)}, x∈ {R}₊², \\ (-{Δ})v(x)= x{ }ᵇ{u}^{{p}₂}(x){e}^{{q}₂v(x)}, x∈ {R}₊², {cases}$ with Dirichlet boundary conditions, where 0 < α < 2 and p 1 , p 2 , q 1 , q 2 > 0. First, we derived the integral representation formula corresponding to the above system under the assumption <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≥</m:mo> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> </m:mfrac> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:math> p₁≥ -2a/α -1 . Then, we prove Liouville theorem for solutions to the above system via the method of scaling spheres.