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• W4220970328 abstract "Abstract For the quasilinear Schrödinger equation <?CDATA begin{equation*}-{Delta}u+V(x)u+frac{kappa }{2}{Delta}({u}^{2})u=h(u),quad uin {H}^{1}({mathbb{R}}^{N}),end{equation*}?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=block overflow=scroll> <mml:mo>−</mml:mo> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>x</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>κ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfrac> <mml:mi mathvariant=normal>Δ</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width=1em /> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:math> where N ⩾ 3, κ is a real parameter, V ( x ) = V (| x |) is a potential allowed to be singular at the origin and <?CDATA $h:mathbb{R}to mathbb{R}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline overflow=scroll> <mml:mi>h</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant=double-struck>R</mml:mi> <mml:mo>→</mml:mo> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:math> is a nonlinearity satisfying conditions similar to those in the paper (1983 Arch. Ration. Mech. Anal. 82 347–375) by Berestycki and Lions, we establish the existence of infinitely many radial solutions for κ < 0 and the existence of more and more radial solutions as κ ↓ 0. In the case κ < 0, we allow h ( u ) = | u | p −2 u for p in the whole range (2, 4 N /( N − 2)) and this is in sharp contrast to most of the existing results which are only for p ∈ [4, 4 N /( N − 2)). Moreover, our result in this case extends the result of Berestycki and Lions in the paper mentioned above to quasilinear equations with singular potentials. In the case κ ⩾ 0, our result extends and covers several related results in the literature, including the result of Berestycki and Lions." @default.
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• W4220970328 date "2022-02-18" @default.
• W4220970328 modified "2023-09-28" @default.
• W4220970328 title "Quasilinear Schrödinger equations involving singular potentials" @default.
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