<p>In this paper, we are concerned with a nonlinear Schrödinger equation as follows: <Equation ID="Equa"> <EquationSource Format="MATHML"><math> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <mi>λ</mi> <mi>u</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mtext> in </mtext> <msup> <mi mathvariant="double-struck">R</mi> <mi>N</mi> </msup> <mo>,</mo> </math></EquationSource> <EquationSource Format="TEX">$ - {\Delta u} + {\lambda u} = g\left ( u\right ) \text{ in }{\mathbb{R}}^{N}, $</EquationSource> </Equation> where <i>g</i> satisfies the Berestycki–Lions conditions. We are going to prove the equation mentioned above has at least <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>κ</mi> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> </math></EquationSource> <EquationSource Format="TEX">$\kappa \left ( N\right )$</EquationSource> </InlineEquation> non-spherically symmetric solutions with <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>κ</mi> <mrow> <mo>(</mo> <mi>N</mi> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> <EquationSource Format="TEX">$\kappa \left ( N\right ) \rightarrow \infty $</EquationSource> </InlineEquation> as <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </math></EquationSource> <EquationSource Format="TEX">$N \rightarrow \infty $</EquationSource> </InlineEquation>. The proof is based on a property of invariant subspace under group action.</p>

Multiple non-spherically symmetric solutions for a nonlinear Schrödinger equation

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2026年2月4日

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Hanqing Zhu,
Xianyong Yang

摘要

In this paper, we are concerned with a nonlinear Schrödinger equation as follows: $- Δ u + λ u = g (u) in R^{N},$ $ - {\Delta u} + {\lambda u} = g\left ( u\right ) \text{ in }{\mathbb{R}}^{N}, $ where g satisfies the Berestycki–Lions conditions. We are going to prove the equation mentioned above has at least $κ (N)$ $\kappa \left ( N\right )$ non-spherically symmetric solutions with $κ (N) \to \infty$ $\kappa \left ( N\right ) \rightarrow \infty $ as $N \to \infty$ $N \rightarrow \infty $ . The proof is based on a property of invariant subspace under group action.