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• W3128705230 abstract "Abstract The purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: $$ Delta ^{2}u- biggl(a+ b int _{mathbb{R}^{N}} vert nabla u vert ^{2},dx biggr) Delta u+V(x)u=K(x)f(u) quadtext{in } mathbb{R}^{N}, $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:msub> <mml:mo>|</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mspace /> <mml:mtext>in </mml:mtext> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:math> where $5leq Nleq 7$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mn>5</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>N</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>7</mml:mn> </mml:math> , $Delta ^{2}$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Δ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> denotes the biharmonic operator, $K(x), V(x)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>K</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> are positive continuous functions which vanish at infinity, and $f(u)$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions." @default.
• W3128705230 created "2021-02-15" @default.
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• W3128705230 date "2021-01-29" @default.
• W3128705230 modified "2023-09-26" @default.
• W3128705230 title "Sign-changing solutions for Schrödinger–Kirchhoff-type fourth-order equation with potential vanishing at infinity" @default.
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• W3128705230 doi "https://doi.org/10.1186/s13660-021-02552-8" @default.
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