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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.11782 |
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| _version_ | 1866929314067382272 |
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| author | Zhang, Yibin |
| author_facet | Zhang, Yibin |
| contents | Given a bounded smooth domain $Ω$ in $\mathbb{R}^2$, we study the following anisotropic elliptic problem $$ \begin{cases} -\nabla\big(a(x)\nabla \upsilon\big)= a(x)\big[e^{\upsilon}-sϕ_1-4παδ_q-h(x)\big]\,\,\,\, \,\textrm{in}\,\,\,\,\,Ω,\\[2mm] \upsilon=0 \qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\quad \textrm{on}\,\ \,\partialΩ, \end{cases} $$ where $a(x)$ is a positive smooth function, $s>0$ is a large parameter, $h\in C^{0,γ}(\overlineΩ)$, $q\inΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $δ_q$ denotes the Dirac measure with pole at point $q$ and $ϕ_1$ is a positive first eigenfunction of the problem $-\nabla\big(a(x)\nabla ϕ\big)=λa(x)ϕ$ under Dirichlet boundary condition in $Ω$. We show that if $q$ is both a local maximum point of $ϕ_1$ and an isolated local maximum point of $a(x)ϕ_1$, this problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $q$ and the quantity $\int_Ωa(x)e^{\upsilon_s}\rightarrow8π(m+1+α)a(q)ϕ_1(q)$ as $s\rightarrow+\infty$, which give a positive answer to the Lazer-McKenna conjecture for this case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11782 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Lazer-McKenna conjecture for an anisotropic planar exponential nonlinearity with a singular source Zhang, Yibin Analysis of PDEs Given a bounded smooth domain $Ω$ in $\mathbb{R}^2$, we study the following anisotropic elliptic problem $$ \begin{cases} -\nabla\big(a(x)\nabla \upsilon\big)= a(x)\big[e^{\upsilon}-sϕ_1-4παδ_q-h(x)\big]\,\,\,\, \,\textrm{in}\,\,\,\,\,Ω,\\[2mm] \upsilon=0 \qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\quad \textrm{on}\,\ \,\partialΩ, \end{cases} $$ where $a(x)$ is a positive smooth function, $s>0$ is a large parameter, $h\in C^{0,γ}(\overlineΩ)$, $q\inΩ$, $α\in(-1,+\infty)\setminus\mathbb{N}$, $δ_q$ denotes the Dirac measure with pole at point $q$ and $ϕ_1$ is a positive first eigenfunction of the problem $-\nabla\big(a(x)\nabla ϕ\big)=λa(x)ϕ$ under Dirichlet boundary condition in $Ω$. We show that if $q$ is both a local maximum point of $ϕ_1$ and an isolated local maximum point of $a(x)ϕ_1$, this problem has a family of solutions $\upsilon_s$ with arbitrary $m$ bubbles accumulating to $q$ and the quantity $\int_Ωa(x)e^{\upsilon_s}\rightarrow8π(m+1+α)a(q)ϕ_1(q)$ as $s\rightarrow+\infty$, which give a positive answer to the Lazer-McKenna conjecture for this case. |
| title | The Lazer-McKenna conjecture for an anisotropic planar exponential nonlinearity with a singular source |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.11782 |