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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.05644 |
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| _version_ | 1866912380307374080 |
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| author | Lopera, Emer Recôva, Leandro Rumbos, Adolfo |
| author_facet | Lopera, Emer Recôva, Leandro Rumbos, Adolfo |
| contents | In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1}
\left\{ \begin{aligned}
-(Δ)_{p}^{s} u + V(x)|u|^{p-2}u &= λf(u),&x\inΩ;
u&=0,&x\in \R^{N}\backslashΩ,
\end{aligned}
\right. \end{equation} where $Ω\subset\R^{N}$ is an open bounded set with Lipschitz boundary $\partialΩ$, $N\geqslant 2,$ $V\in L^{\infty}(\R^{N})$, and $(-Δ)_p^s$ denotes the fractional $p$-Laplacian with $s\in(0,1), 1<p$, $sp<N$, $λ>0$, and $f:\R\rightarrow\R$ is a continuous function. We extend the results of Lopera {\it et al.} in \cite{Lopera1} by proving the existence of a second weak solution for problem (\ref{probaa1}). We apply a variant of the mountain-pass theorem due to Hofer \cite{Hofer2} and infinite-dimensional Morse theory to obtain the existence of at least two solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05644 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiplicity results for Schrödinger type fractional $p$-Laplacian boundary value problems Lopera, Emer Recôva, Leandro Rumbos, Adolfo Analysis of PDEs 35J20 In this work, we study the existence and multiplicity of solutions for the following problem \begin{equation}\label{probaa1} \left\{ \begin{aligned} -(Δ)_{p}^{s} u + V(x)|u|^{p-2}u &= λf(u),&x\inΩ; u&=0,&x\in \R^{N}\backslashΩ, \end{aligned} \right. \end{equation} where $Ω\subset\R^{N}$ is an open bounded set with Lipschitz boundary $\partialΩ$, $N\geqslant 2,$ $V\in L^{\infty}(\R^{N})$, and $(-Δ)_p^s$ denotes the fractional $p$-Laplacian with $s\in(0,1), 1<p$, $sp<N$, $λ>0$, and $f:\R\rightarrow\R$ is a continuous function. We extend the results of Lopera {\it et al.} in \cite{Lopera1} by proving the existence of a second weak solution for problem (\ref{probaa1}). We apply a variant of the mountain-pass theorem due to Hofer \cite{Hofer2} and infinite-dimensional Morse theory to obtain the existence of at least two solutions. |
| title | Multiplicity results for Schrödinger type fractional $p$-Laplacian boundary value problems |
| topic | Analysis of PDEs 35J20 |
| url | https://arxiv.org/abs/2408.05644 |