<p>In this paper, we investigate the existence of normalized solutions for a quasilinear elliptic problem as follows <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_3547_Article_Equ1.gif" Format="GIF" Height="54" Rendition="HTML" Resolution="72" Type="Linedraw" Width="279" /> </MediaObject> <EquationSource Format="TEX">\(\begin{cases}\Delta_{p}u+\lambda u^{p-1}=f(u), &amp; x \in \mathbb{R}^{N},\\\int_{\mathbb{R}^{N}}|u|^{p}dx=\rho, &amp; u \in W^{1,p}(\mathbb{R}^{N})\end{cases}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>{</mo> <mtable columnalign="left left" columnspacing="1em" displaystyle="false" rowspacing=".2em"> <mtr> <mtd> <msub> <mi mathvariant="normal">Δ</mi> <mrow> <mi>p</mi> </mrow> </msub> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <msup> <mi>u</mi> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> <mtd> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mo>∫</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>u</mi> <msup> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> </mrow> </msup> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>ρ</mi> <mo>,</mo> </mtd> <mtd> <mi>u</mi> <mo>∈</mo> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>N</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" /> </mrow> </math></EquationSource> </Equation> where −Δ<sub><i>p</i></sub> is the <i>p</i>-Laplace operator, 1 &lt; <i>p</i> &lt; <i>N, N</i> ≥ 3, <i>ρ</i> &gt; 0 and λ &gt; 0. <i>f</i> is a continuous function and satisfies some suitable conditions. Based on a Nehari–Pohozaev manifold, we show the existence of positive normalized solutions by using the minimization method.</p>

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Existence of Normalized Ground State Solutions for Quasilinear Elliptic Problems in ℝN

  • Yuanyuan Li,
  • Jingbo Dou

摘要

In this paper, we investigate the existence of normalized solutions for a quasilinear elliptic problem as follows \(\begin{cases}\Delta_{p}u+\lambda u^{p-1}=f(u), & x \in \mathbb{R}^{N},\\\int_{\mathbb{R}^{N}}|u|^{p}dx=\rho, & u \in W^{1,p}(\mathbb{R}^{N})\end{cases}\) { Δ p u + λ u p 1 = f ( u ) , x R N , R N | u | p d x = ρ , u W 1 , p ( R N ) where −Δp is the p-Laplace operator, 1 < p < N, N ≥ 3, ρ > 0 and λ > 0. f is a continuous function and satisfies some suitable conditions. Based on a Nehari–Pohozaev manifold, we show the existence of positive normalized solutions by using the minimization method.