<p>In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation <Equation ID="Equ62"> <EquationSource Format="TEX">$\begin{aligned} -\nabla \cdot \left( |x|^{2a} \nabla u\right) + \omega u=|u|^{p-2}u \quad \text{ in } \,\, \mathbb {R}^d, \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mfenced close=")" open="("> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>u</mi> </mfenced> <mo>+</mo> <mi>ω</mi> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mspace width="0.166667em" /> <mspace width="0.166667em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d \ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<a<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>a</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\omega >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2<p<\frac{2d}{d-2(1-a)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mfrac> <mrow> <mn>2</mn> <mi>d</mi> </mrow> <mrow> <mi>d</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. We proved that any ground state is radially symmetric and strictly decreasing in the radial direction. Moreover, we establish the uniqueness of ground states and derive the non-degeneracy of ground states in the corresponding radially symmetric Sobolev space. This affirms the natural conjectures posed recently in Iyer and Stefanov (Calc. Var. Partial. Differ. Equ. <b>64</b>:19, 2025).</p>

Radial symmetry, uniqueness and non-degeneracy of solutions to degenerate nonlinear Schrödinger equations

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发表日期：

2026年3月14日

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Tianxiang Gou

摘要

In this paper, we consider the radial symmetry, uniqueness and non-degeneracy of solutions to the degenerate nonlinear elliptic equation $\begin{aligned} -\nabla \cdot \left( |x|^{2a} \nabla u\right) + \omega u=|u|^{p-2}u \quad \text{ in } \,\, \mathbb {R}^d, \end{aligned}$ $\begin{matrix} - \nabla \cdot ({| x |}^{2 a}, \nabla, u) + ω u = {| u |}^{p - 2} u in R^{d}, \end{matrix}$ where $d \ge 2$ $d \geq 2$ , $0<a<1$ $0 < a < 1$ , $\omega >0$ $ω > 0$ and $2<p<\frac{2d}{d-2(1-a)}$ $2 < p < \frac{2 d}{d - 2 (1 - a)}$ . We proved that any ground state is radially symmetric and strictly decreasing in the radial direction. Moreover, we establish the uniqueness of ground states and derive the non-degeneracy of ground states in the corresponding radially symmetric Sobolev space. This affirms the natural conjectures posed recently in Iyer and Stefanov (Calc. Var. Partial. Differ. Equ. 64:19, 2025).