On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity
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| Format: | Preprint |
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2024
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| _version_ | 1866910743374331904 |
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| author | Saldaña, Alberto Schiera, Delia Tavares, Hugo |
| author_facet | Saldaña, Alberto Schiera, Delia Tavares, Hugo |
| contents | We consider the following Lane-Emden system with Neumann boundary conditions \[ -Δu= |v|^{q-1}v \text{ in } Ω,\qquad -Δv= |u|^{p-1}u \text{ in } Ω,\qquad \partial_νu=\partial_νv=0 \text{ on } \partial Ω, \] where $Ω$ is a bounded smooth domain of $\mathbb{R}^N$ with $N \ge 1$. We study the multiplicity of solutions and the convergence of least energy (nodal) solutions (l.e.s.) as the exponents $p, q > 0$ vary in the subcritical regime $1/(p+1) + 1/(q+1) > (N-2)/N$, or in the critical case $1/(p+1) + 1/(q+1) =(N-2)/N$ with some additional assumptions. We consider, for the first time in this setting, the cases where one or two exponents tend to zero, proving that l.e.s. converge to a problem with a sign nonlinearity. Our approach is based on an alternative characterization of least energy levels in terms of the nonlinear eigenvalue problem \[ Δ(|Δu|^{\frac 1 q -1} Δu) = λ|u|^{p-1} u, \quad \partial_νu=\partial_ν(|Δu|^{\frac 1 q -1} Δu)=0 \text{ on } \partial Ω. \] As an application, we show a symmetry breaking phenomenon for l.e.s. of a bilaplacian equation with sign nonlinearity and for other equations with nonlinear higher-order operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_09512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity Saldaña, Alberto Schiera, Delia Tavares, Hugo Analysis of PDEs We consider the following Lane-Emden system with Neumann boundary conditions \[ -Δu= |v|^{q-1}v \text{ in } Ω,\qquad -Δv= |u|^{p-1}u \text{ in } Ω,\qquad \partial_νu=\partial_νv=0 \text{ on } \partial Ω, \] where $Ω$ is a bounded smooth domain of $\mathbb{R}^N$ with $N \ge 1$. We study the multiplicity of solutions and the convergence of least energy (nodal) solutions (l.e.s.) as the exponents $p, q > 0$ vary in the subcritical regime $1/(p+1) + 1/(q+1) > (N-2)/N$, or in the critical case $1/(p+1) + 1/(q+1) =(N-2)/N$ with some additional assumptions. We consider, for the first time in this setting, the cases where one or two exponents tend to zero, proving that l.e.s. converge to a problem with a sign nonlinearity. Our approach is based on an alternative characterization of least energy levels in terms of the nonlinear eigenvalue problem \[ Δ(|Δu|^{\frac 1 q -1} Δu) = λ|u|^{p-1} u, \quad \partial_νu=\partial_ν(|Δu|^{\frac 1 q -1} Δu)=0 \text{ on } \partial Ω. \] As an application, we show a symmetry breaking phenomenon for l.e.s. of a bilaplacian equation with sign nonlinearity and for other equations with nonlinear higher-order operators. |
| title | On least energy solutions to a pure Neumann Lane-Emden system: convergence, symmetry breaking, and multiplicity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.09512 |