Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials

The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:munder> <m:mrow> <m:mi>sup</m:mi> </m:mrow> <m:mrow> <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:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>∇</m:mi> <m:mi>u</m:mi> <m:msup> <m:mrow> <m:mo stretchy="false">|</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 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>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:munder> <m:munder> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:munder> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mn>4</m:mn> <m:mi>π</m:mi> <m:mfenced close=")" open="("> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mi>β</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:mrow> </m:msup> <m:mo>−</m:mo> <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>β</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mi mathvariant="normal">d</m:mi> <m:mi>x</m:mi> <m:mo><</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> <m:mo>,</m:mo> </m:math> $${u∈ {H}¹({R}²),{{R}²}{∫ }( ∇ u{ }²+V(x){u}²)dx≤ 1}{sup}{{R}²}{∫ }{{e}^{4π (1-β /2){u}²}-1}{ x{ }β}dx{< }+∞ ,$$ where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>β</m:mi> <m:mo><</m:mo> <m:mn>2</m:mn> </m:math> $0{< }β {< }2$ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <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:mn>0</m:mn> </m:math> V(x)≥ 0 and may vanish on an open set in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> R² . Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> R² : (0.2) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mo>−</m:mo> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>γ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </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>β</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>,</m:mo> </m:math> -Δu+γ u=f(u) x β, where the nonlinearity <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>f</m:mi> </m:math> f has the critical exponential growth. In order to overcome the lack of compactness, we develop a method which is based on the threshold of the least energy, an embedding theorem introduced in (C. Zhang and L. Chen, “Concentration-compactness principle of singular Trudinger-Moser inequalities in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <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:math> Rⁿ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>n</m:mi> </m:math> n -Laplace equations,” Adv. Nonlinear Stud. , vol. 18, no. 3, pp. 567–585, 2018) and the Nehari manifold to get the existence of ground states. Furthermore, as an application of inequality (0.1), we also prove the existence of ground states to the following equations involving degenerate potentials in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> R² : (0.3) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <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:mi>x</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mfrac> <m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </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>β</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mo>.</m:mo> </m:math> -Δu+V(x)u=f(u) x β.