Sharp Trudinger–Moser Inequality and Ground State Solutions to Quasi-Linear Schrödinger Equations with Degenerate Potentials in ℝ^<i>n</i>

Abstract

Abstract The main purpose of this paper is to establish the existence of ground-state solutions to a class of Schrödinger equations with critical exponential growth involving the nonnegative, possibly degenerate, potential V: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>div</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo>⁢</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>⁢</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> -\operatorname{div}(\lvert\nabla u\rvert^{n-2}\nabla u)+V(x)\lvert u\rvert^{n-% 2}u=f(u). To this end, we first need to prove a sharp Trudinger–Moser inequality in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} under the constraint <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msup> <m:mo>+</m:mo> <m:mrow> <m:mi>V</m:mi> <m:mo>⁢</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>⁢</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:mo fence="true" stretchy="false">|</m:mo> </m:mrow> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> <jats:graphic xmlns:

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