Global solution curves in harmonic parameters, and multiplicity of solutions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908779232100352 |
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| author | Korman, Philip |
| author_facet | Korman, Philip |
| contents | \[ Δu+g(u)=f(x) \s \mbox{for $x \in Ω$}, \s u=0 \s \mbox{on $\partial Ω$} \] decompose $f(x)=μ_1 \p _1+e(x)$, where $\p _1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x) \perp \p _1$ in $L^2(Ω)$, and similarly write $u(x)= ξ_1 \p _i+U (x)$, with $ U \perp \p _1$ in $L^2(Ω)$. We study properties of the solution curve $(u(x),μ_1)(ξ_1)$, and in particular its section $μ_1=μ_1(ξ_1)$, which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption $g'(u)<\la _2$. We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_14581 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global solution curves in harmonic parameters, and multiplicity of solutions Korman, Philip Analysis of PDEs Classical Analysis and ODEs Dynamical Systems \[ Δu+g(u)=f(x) \s \mbox{for $x \in Ω$}, \s u=0 \s \mbox{on $\partial Ω$} \] decompose $f(x)=μ_1 \p _1+e(x)$, where $\p _1$ is the principal eigenfunction of the Laplacian with zero boundary conditions, and $e(x) \perp \p _1$ in $L^2(Ω)$, and similarly write $u(x)= ξ_1 \p _i+U (x)$, with $ U \perp \p _1$ in $L^2(Ω)$. We study properties of the solution curve $(u(x),μ_1)(ξ_1)$, and in particular its section $μ_1=μ_1(ξ_1)$, which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption $g'(u)<\la _2$. We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results. |
| title | Global solution curves in harmonic parameters, and multiplicity of solutions |
| topic | Analysis of PDEs Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2601.14581 |