On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities
We deal with existence and multiplicity for the following class of nonhomogeneous Schrödinger–Poisson systems \begin{equation*} \begin{cases} -\Delta u + V(x) u + K(x) \phi(x) u = f(x, u) + g(x) \quad & \text{in } \mathbb{R}^3, \\ -\Delta \phi = K(x) u^2 \quad & \text{in } \mathbb{R}^...
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| Format: | Article |
| Language: | English |
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University of Szeged
2017-04-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5492 |
| _version_ | 1797830561420541952 |
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| author | Sara Barile |
| author_facet | Sara Barile |
| author_sort | Sara Barile |
| collection | DOAJ |
| description | We deal with existence and multiplicity for the following class of nonhomogeneous Schrödinger–Poisson systems
\begin{equation*}
\begin{cases}
-\Delta u + V(x) u + K(x) \phi(x) u = f(x, u) + g(x) \quad & \text{in } \mathbb{R}^3, \\
-\Delta \phi = K(x) u^2 \quad & \text{in } \mathbb{R}^3,
\end{cases}
\end{equation*}
where $V, K: \mathbb{R}^3 \rightarrow \mathbb{R}^+$ are suitable potentials and $f: \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}$ satisfies sublinear growth assumptions involving a finite number of positive weights $W_i$, $i= 1,\dots,r$ with $r \geq 1$. By exploiting compact embeddings of the functional space on which we work in every weighted space $L_{W_i}^{w_i}(\mathbb{R}^3)$, $w_i \in (1, 2)$, we establish existence by means of a generalized Weierstrass theorem. Moreover, we prove multiplicity of solutions if $f$ is odd in $u$ and $g(x) \equiv 0$ thanks to a variant of the symmetric mountain pass theorem stated by R. Kajikiya for subquadratic functionals. |
| first_indexed | 2024-04-09T13:38:03Z |
| format | Article |
| id | doaj.art-24a6631f9c764697a986a432272a3db5 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:03Z |
| publishDate | 2017-04-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-24a6631f9c764697a986a432272a3db52023-05-09T07:53:07ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752017-04-0120172112110.14232/ejqtde.2017.1.215492On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearitiesSara Barile0Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Bari, ItalyWe deal with existence and multiplicity for the following class of nonhomogeneous Schrödinger–Poisson systems \begin{equation*} \begin{cases} -\Delta u + V(x) u + K(x) \phi(x) u = f(x, u) + g(x) \quad & \text{in } \mathbb{R}^3, \\ -\Delta \phi = K(x) u^2 \quad & \text{in } \mathbb{R}^3, \end{cases} \end{equation*} where $V, K: \mathbb{R}^3 \rightarrow \mathbb{R}^+$ are suitable potentials and $f: \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{R}$ satisfies sublinear growth assumptions involving a finite number of positive weights $W_i$, $i= 1,\dots,r$ with $r \geq 1$. By exploiting compact embeddings of the functional space on which we work in every weighted space $L_{W_i}^{w_i}(\mathbb{R}^3)$, $w_i \in (1, 2)$, we establish existence by means of a generalized Weierstrass theorem. Moreover, we prove multiplicity of solutions if $f$ is odd in $u$ and $g(x) \equiv 0$ thanks to a variant of the symmetric mountain pass theorem stated by R. Kajikiya for subquadratic functionals.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5492schrödinger–poisson systemssublinear nonlinearitiesvariational methodscompact embeddings |
| spellingShingle | Sara Barile On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities Electronic Journal of Qualitative Theory of Differential Equations schrödinger–poisson systems sublinear nonlinearities variational methods compact embeddings |
| title | On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities |
| title_full | On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities |
| title_fullStr | On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities |
| title_full_unstemmed | On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities |
| title_short | On existence and multiplicity for Schrödinger–Poisson systems involving weighted sublinear nonlinearities |
| title_sort | on existence and multiplicity for schrodinger poisson systems involving weighted sublinear nonlinearities |
| topic | schrödinger–poisson systems sublinear nonlinearities variational methods compact embeddings |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5492 |
| work_keys_str_mv | AT sarabarile onexistenceandmultiplicityforschrodingerpoissonsystemsinvolvingweightedsublinearnonlinearities |