Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909746704941056 |
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| author | Clapp, Mónica Culebro, Carlos |
| author_facet | Clapp, Mónica Culebro, Carlos |
| contents | We consider the nonlinear elliptic equation \begin{equation*} -Δu + V(x)u = f(u), \qquad u\in D^{1,2}_0(Ω), \end{equation*} in an exterior domain $Ω$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at infinity and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions $N\geq 4$ there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity Clapp, Mónica Culebro, Carlos Analysis of PDEs 2010: 35Q55, 35B06, 35J20 We consider the nonlinear elliptic equation \begin{equation*} -Δu + V(x)u = f(u), \qquad u\in D^{1,2}_0(Ω), \end{equation*} in an exterior domain $Ω$ of $\mathbb{R}^N$, where $V$ is a scalar potential that decays to zero at infinity and the nonlinearity $f$ is subcritical at infinity and supercritical near the origin. Under weak symmetry assumptions, we provide conditions that guarantee that this problem has a prescribed number of sign-changing solutions. In particular, we show that in dimensions $N\geq 4$ there are numerous examples of exterior domains with finite symmetries in which the problem has a predetermined number of nodal solutions. |
| title | Multiple nodal solutions to a scalar field equation with double-power nonlinearity and zero mass at infinity |
| topic | Analysis of PDEs 2010: 35Q55, 35B06, 35J20 |
| url | https://arxiv.org/abs/2508.15167 |