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• W2890965148 abstract "Abstract We study the following generalized quasilinear Schrödinger equation: <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:msup> <m:mi>g</m:mi> <m:mn>2</m:mn> </m:msup> <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: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:mo>+</m:mo> <m:mrow> <m:mi>g</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:mo>⁢</m:mo> <m:msup> <m:mi>g</m:mi> <m:mo>′</m:mo> </m:msup> <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:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy=false>|</m:mo> <m:mrow> <m:mo>∇</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=false>|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </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:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi>h</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 rspace=12.5pt>,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> -(g^{2}(u)nabla u)+g(u)g^{prime}(u)|nabla u|^{2}+V(x)u=h(u),quad xin% mathbb{R}^{N}, where <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>N</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> {Ngeq 3} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>g</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>ℝ</m:mi> <m:mo>→</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msup> </m:mrow> </m:mrow> </m:math> {gcolonmathbb{R}rightarrowmathbb{R}^{+}} is an even differentiable function such that <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi>g</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>t</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {g^{prime}(t)geq 0} for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>t</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {tgeq 0} , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>h</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℝ</m:mi> <m:mo>,</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {hin C^{1}(mathbb{R},mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <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:mrow> <m:mo>:</m:mo> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {V(x)colonmathbb{R}^{N}rightarrowmathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on <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> {mathbf{R}}^{N} , Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem." @default.
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• W2890965148 title "Ground State Solutions for Quasilinear Schrödinger Equations with Critical Growth and Lower Power Subcritical Perturbation" @default.
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