Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914233421135872 |
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| author | Vitillaro, Enzo |
| author_facet | Vitillaro, Enzo |
| contents | The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} Δu=0 \qquad &\text{in $Ω$,}\\ u=0 &\text{on $Γ_0$,}\\ -Δ_Γu +\partial_νu =|u|^{p-2}u\qquad &\text{on $Γ_1$,} \end{cases} $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\partialΩ=Γ_0\cupΓ_1$, $Γ_0\capΓ_1=\emptyset$, $Γ_1$ being nonempty and relatively open on $Γ$, $\mathcal{H}^{N-1}(Γ_0)>0$ and $p>2$ being subcritical with respect to Sobolev embedding on $\partialΩ$.
We prove that the problem admits nontrivial solutions at the potential--well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_06442 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition Vitillaro, Enzo Analysis of PDEs 35D30, 35J05, 35J20, 25J25, 35J61, 35J67 The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} Δu=0 \qquad &\text{in $Ω$,}\\ u=0 &\text{on $Γ_0$,}\\ -Δ_Γu +\partial_νu =|u|^{p-2}u\qquad &\text{on $Γ_1$,} \end{cases} $$ where $Ω$ is a bounded open subset of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\partialΩ=Γ_0\cupΓ_1$, $Γ_0\capΓ_1=\emptyset$, $Γ_1$ being nonempty and relatively open on $Γ$, $\mathcal{H}^{N-1}(Γ_0)>0$ and $p>2$ being subcritical with respect to Sobolev embedding on $\partialΩ$. We prove that the problem admits nontrivial solutions at the potential--well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels. |
| title | Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition |
| topic | Analysis of PDEs 35D30, 35J05, 35J20, 25J25, 35J61, 35J67 |
| url | https://arxiv.org/abs/2310.06442 |