Infinitely many new multi-bumps solutions for the nonlinear Schrödinger equation

In this paper we consider the existence of novel multi-bumps solutions for the following Schrödinger equation (0.1) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>V</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mi>u</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> <m:mspace width="0.17em"/> <m:mspace width="0.3333em"/> <m:mtext>in</m:mtext> <m:mspace width="0.17em"/> <m:mspace width="0.3333em"/> <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:mspace width="0.17em"/> <m:mspace width="0.3333em"/> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <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:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> -Δu+V( x )u=uᵖ, u >0 in RN, u∈ H¹(RN), where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mfrac> <m:mrow> <m:mi>N</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mrow> <m:mi>N</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> p∈ (1,N+2/N-2) and V is positive radial function satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mi>V</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <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>n</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>+</m:mo> <m:mi>O</m:mi> <m:mfenced open="(" close=")"> <m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <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>n</m:mi> <m:mo>+</m:mo> <m:mi>θ</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> <m:mo>,</m:mo> <m:mspace width="0.17em"/> <m:mspace width="0.3333em"/> <m:mtext>as</m:mtext> <m:mspace width="0.17em"/> <m:mspace width="0.3333em"/> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> <m:mo>→</m:mo> <m:mi>∞</m:mi> <m:mo>,</m:mo> </m:math> V( x )=V₀+a x ⁿ+O(1 x n+θ), as x → ∞ , with V 0