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• W2964136454 abstract "Abstract In this paper, we study the following biharmonic equations: <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing=0pt displaystyle=true rowspacing=0pt> <m:mtr> <m:mtd /> <m:mtd columnalign=left> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi mathvariant=normal>Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mi mathvariant=normal>Δ</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mrow> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mi>b</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:msub> <m:mi>b</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <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>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:mtd> <m:mtd columnalign=left> <m:mrow> <m:mrow> <m:mpadded lspace=10pt width=+10pt> <m:mtext>in </m:mtext> </m:mpadded> <m:mo>⁢</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd /> <m:mtd columnalign=left> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> $left{begin{aligned} &displaystyleDelta^{2}u-a_{0}Delta u+(lambda b(x)+b% _{0})u=f(u)&hskip 10.0pttext{in }mathbb{R}^{N}, &displaystyle uin H^{2}(mathbb{R}^{N}),end{aligned}right.$ 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:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>b</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> <m:mo>∈</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> ${a_{0},b_{0}inmathbb{R}}$ are two constants, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>λ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${lambda>0}$ is a parameter, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>b</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:mn>0</m:mn> </m:mrow> </m:math> ${b(x)geq 0}$ is a potential well and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:mi>f</m:mi> <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:mrow> <m:mi>C</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:mi>ℝ</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> ${f(t)in C(mathbb{R})}$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo–Nirenberg inequality, we make some observations on the operator <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mrow> <m:msup> <m:mi mathvariant=normal>Δ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>-</m:mo> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mi mathvariant=normal>Δ</m:mi> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi>λ</m:mi> <m:mo>⁢</m:mo> <m:mi>b</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:msub> <m:mi>b</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> ${Delta^{2}-a_{0}Delta+lambda b(x)+b_{0}}$ in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=false>(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mrow> </m:math> ${H^{2}(mathbb{R}^{N})}$ . Based on these observations, we give a new variational setting to the above problem for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo><</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${a_{0}<0}$ . With this new variational setting in hands, we establish some new existence results of the nontrivial solutions for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo><</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${a_{0}<0}$ with λ sufficiently large by the variational method. The concentration behavior of the nontrivial solution for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mi>λ</m:mi> <m:mo>→</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant=normal>∞</m:mi> </m:mrow> </m:mrow> </m:math> ${lambdato+infty}$ is also obtained. It is worth pointing out that it seems to be the first time that the nontrivial solution of the above problem is obtained for all <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:msub> <m:mi>a</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo><</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> ${a_{0}<0}$ ." @default.
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• W2964136454 date "2016-10-11" @default.
• W2964136454 modified "2023-09-23" @default.
• W2964136454 title "On a Biharmonic Equation with Steep Potential Well and Indefinite Potential" @default.
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