Multiplicity of solutions to a degenerate elliptic equation in the sub-critical and critical cases
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909417911353344 |
|---|---|
| author | Bal, Kaushik Biswas, Sanjit |
| author_facet | Bal, Kaushik Biswas, Sanjit |
| contents | Given a smooth and bounded domain $Ω(\subset\mathbf{R}^N)$, we prove the existence of two non-trivial, non-negative solutions for the semilinear degenerate elliptic equation \begin{align} \left. \begin{array}{l} -Δ_λu=μg(z)|u|^{r-1}u+h(z)|u|^{s-1}u \;\text{in}\; Ωu\in H^{1,λ}_0(Ω) \end{array}\right\} \end{align} where $Δ_λ=Δ_x+|x|^{2λ}Δ_y$ is the Grushin Laplacian Operator, $z=(x,y)\inΩ$, $N=n+m;\, n,\, m\geq 1$, $λ>0$, $0\leq r<1<s<2^*_λ-1$; the functions $g,h$ are of indefinite sign and $2^*_λ=\frac{2Q}{Q-2}$ is the critical Sobolev exponent, where $Q=n+(1+λ)m$ is the homogeneous dimension associated to the operator $Δ_λ$. As for the critical case $s=2^*_λ-1$, we prove the existence of at least two non-trivial, non-negative solutions provided $g\geq 0$ and $h\equiv 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_04794 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicity of solutions to a degenerate elliptic equation in the sub-critical and critical cases Bal, Kaushik Biswas, Sanjit Analysis of PDEs Given a smooth and bounded domain $Ω(\subset\mathbf{R}^N)$, we prove the existence of two non-trivial, non-negative solutions for the semilinear degenerate elliptic equation \begin{align} \left. \begin{array}{l} -Δ_λu=μg(z)|u|^{r-1}u+h(z)|u|^{s-1}u \;\text{in}\; Ωu\in H^{1,λ}_0(Ω) \end{array}\right\} \end{align} where $Δ_λ=Δ_x+|x|^{2λ}Δ_y$ is the Grushin Laplacian Operator, $z=(x,y)\inΩ$, $N=n+m;\, n,\, m\geq 1$, $λ>0$, $0\leq r<1<s<2^*_λ-1$; the functions $g,h$ are of indefinite sign and $2^*_λ=\frac{2Q}{Q-2}$ is the critical Sobolev exponent, where $Q=n+(1+λ)m$ is the homogeneous dimension associated to the operator $Δ_λ$. As for the critical case $s=2^*_λ-1$, we prove the existence of at least two non-trivial, non-negative solutions provided $g\geq 0$ and $h\equiv 1$. |
| title | Multiplicity of solutions to a degenerate elliptic equation in the sub-critical and critical cases |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.04794 |