Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem
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arXiv
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| Format: | Preprint |
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2025
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| author | Alarcón, Salomón Faya, Jorge Rey, Carolina |
| author_facet | Alarcón, Salomón Faya, Jorge Rey, Carolina |
| contents | We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll}
Δ(a(x)Δu) = a(x) \left\vert u \right\vert^{p-2-ε} u \ \text{ in } \ Ω,
\hspace{0.6cm} u = 0 \ \text{ on } \ \partial Ω,
\hspace{0.6cm} Δu = 0 \ \text{ on } \ \partial Ω,
\end{array}\end{equation*} where $Ω$ is a smooth, bounded domain in $\mathbb{R}^N$ with $N \geq 5$. Here, $p := \frac{2N}{N-4}$ is the Sobolev critical exponent for the embedding $H^2 \cap H_0^1(Ω) \hookrightarrow L^p(Ω)$, and $a \in C^2(\overlineΩ)$ is a strictly positive function on $\overlineΩ$.
We establish sufficient conditions on the function $a$ and the domain $Ω$ for this problem to admit both positive and sign-changing solutions with an explicit asymptotic profile. These solutions concentrate and blow up at a point on the boundary $\partial Ω$ as $ε\to 0$. The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_02745 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem Alarcón, Salomón Faya, Jorge Rey, Carolina Analysis of PDEs 35J20, 35J35, 35J60 We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} Δ(a(x)Δu) = a(x) \left\vert u \right\vert^{p-2-ε} u \ \text{ in } \ Ω, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial Ω, \hspace{0.6cm} Δu = 0 \ \text{ on } \ \partial Ω, \end{array}\end{equation*} where $Ω$ is a smooth, bounded domain in $\mathbb{R}^N$ with $N \geq 5$. Here, $p := \frac{2N}{N-4}$ is the Sobolev critical exponent for the embedding $H^2 \cap H_0^1(Ω) \hookrightarrow L^p(Ω)$, and $a \in C^2(\overlineΩ)$ is a strictly positive function on $\overlineΩ$. We establish sufficient conditions on the function $a$ and the domain $Ω$ for this problem to admit both positive and sign-changing solutions with an explicit asymptotic profile. These solutions concentrate and blow up at a point on the boundary $\partial Ω$ as $ε\to 0$. The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method. |
| title | Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem |
| topic | Analysis of PDEs 35J20, 35J35, 35J60 |
| url | https://arxiv.org/abs/2502.02745 |