Properties of ground states of attractive Gross–Pitaevskii equations with multi-well potentials

Abstract

Abstract We are interested in the attractive Gross–Pitaevskii (GP) equation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mstyle> </mml:math> , where the external potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> vanishes on m disjoint bounded domains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mo mathvariant="normal">Ω</mml:mo> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mtext> </mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mstyle> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mo stretchy="false">|</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">|</mml:mo> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mstyle> </mml:math> , that is, the union of these <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mo mathvariant="normal">Ω</mml:mo> <mml:mi>i</mml:mi> </mml:msub> </mml:mstyle> </mml:math> is the bottom of the potential well. By establishing some delicate estimates on the associated energy functional of the GP equation, we prove that when the interaction strength a approaches some critical value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msup> <mml:mi>a</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mstyle> </mml:math> , the ground states concentrate and blow up at the center of the incircle of some <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mo mathvariant="normal">Ω</mml:mo> <mml:mi>j</mml:mi> </mml:msub> </mml:mstyle> </mml:math> with the largest inradius. Moreover, under some further conditions on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> , we show that the ground states of the GP equations are unique and radially symmetric at least for almost every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>a</mml:mi> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mstyle> </mml:math> .