Ground states of nonlocal elliptic equations with general nonlinearities via Rayleigh quotient
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913490996822016 |
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| author | Ferraz, Diego Silva, Edcarlos D. |
| author_facet | Ferraz, Diego Silva, Edcarlos D. |
| contents | It is established ground states and multiplicity of solutions for a nonlocal Schrödinger equation
$(-Δ)^s u + V(x) u = λa(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0<s<\min\{1,N/2\},$ $1<q<2$ and $λ>0,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp $λ^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $λ\in (0, λ^*)$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02205 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ground states of nonlocal elliptic equations with general nonlinearities via Rayleigh quotient Ferraz, Diego Silva, Edcarlos D. Analysis of PDEs It is established ground states and multiplicity of solutions for a nonlocal Schrödinger equation $(-Δ)^s u + V(x) u = λa(x) |u|^{q-2}u + b(x)f(u)$ in $\mathbb{R}^N,$ $u \in H^s(\mathbb{R}^N),$ where $0<s<\min\{1,N/2\},$ $1<q<2$ and $λ>0,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp $λ^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $λ\in (0, λ^*)$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered. |
| title | Ground states of nonlocal elliptic equations with general nonlinearities via Rayleigh quotient |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.02205 |