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<p>We are interested in the multiplicity of solutions to the following scalar field equation <Equation ID="Equ9"> <EquationSource Format="TEX">\( -\varDelta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \text{ in } \mathbb {R}^N \setminus \{0\}. \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>-</mo> <mi>Δ</mi> <mi>u</mi> <mo>-</mo> <mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mn>4</mn> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mfrac> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="0.333333em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We establish the existence of infinitely many radial and nonradial solutions.</p>

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Note on the multiplicity of solutions for nonlinear scalar field equations with a critical inverse-square potential

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  • 发表日期：

    2026年5月25日

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  • Bartosz Bieganowski,
  • Daniel Strzelecki

摘要

We are interested in the multiplicity of solutions to the following scalar field equation \( -\varDelta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \text{ in } \mathbb {R}^N \setminus \{0\}. \) - Δ u - ( N - 2 ) 2 4 | x | 2 u = g ( u ) , in R N \ { 0 } . We establish the existence of infinitely many radial and nonradial solutions.