Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth

In this study, we are interested in multiplicity results for positive solutions of the generalized quasilinear Schrödinger equations with critical growth <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mo>−</m:mo> <m:mi>div</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi>g</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>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>g</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>g</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">′</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mrow> <m:mo>∇</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ε</m:mi> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:mi>p</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ε</m:mi> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mo>∣</m:mo> <m:mi>u</m:mi> <m:mo>∣</m:mo> </m:mrow> <m:mrow> <m:mi>α</m:mi> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mspace width="1.0em"/> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> </m:mrow> </m:math> -div({g}²(u)∇ u)+g(u){g}^{^{} }(u){| ∇ u| }²+V(ε x)u={| u| }α p-2u+Q(ε x){| u| }^{α {2}*-2}u,{1.0em}x∈ {{R}}N, where <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>C</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mo>+</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> g∈ {C}¹({R},{{R}}⁺) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo form="prefix">,</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:math> α ∈ [1,2] , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>2</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:msup> <m:mrow> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:math> 2< p< {2}* , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ε</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> ε > 0 is a parameter. Under suitable assumptions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Q</m:mi> </m:math> Q , we obtain the concentration behavior of positive solutions for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ε</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> ε > 0 small and establish the relationship between the number of positive solutions and the profiles of potentials <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Q</m:mi> </m:math> Q using variational methods.