Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory
In this paper, we deal with the following $p(x)$-Schrödinger problem: \begin{equation*} \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u) & \hbox{in $\mathbb{R}^{N}$ ;} \\ u\in W^{1,p(x)}(\mathbb{R}^{N}), & \hbox{} \end{cases} \end{eq...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2017-11-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5912 |
| _version_ | 1797830579840876544 |
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| author | Rabil Ayazoglu (Mashiyev) Ismail Ekincioglu Gulizar Alisoy |
| author_facet | Rabil Ayazoglu (Mashiyev) Ismail Ekincioglu Gulizar Alisoy |
| author_sort | Rabil Ayazoglu (Mashiyev) |
| collection | DOAJ |
| description | In this paper, we deal with the following $p(x)$-Schrödinger problem:
\begin{equation*}
\begin{cases}
-\text{div}(|\nabla u|^{p(x)-2}\nabla u)+V(x)\left\vert u\right\vert
^{p(x)-2}u=f(x,u) & \hbox{in $\mathbb{R}^{N}$ ;} \\
u\in W^{1,p(x)}(\mathbb{R}^{N}), & \hbox{}
\end{cases}
\end{equation*}
where the nonlinearity is sublinear. We present the existence of infinitely many solutions for the problem. The main tool used here is a variational method and Krasnoselskii's genus theory combined with the theory of variable exponent Sobolev spaces. We also establish a Bartsch–Wang type compact embedding theorem for the variable exponent spaces. |
| first_indexed | 2024-04-09T13:38:21Z |
| format | Article |
| id | doaj.art-62414cc81f784cc5838f10f7c50e3ab3 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:21Z |
| publishDate | 2017-11-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-62414cc81f784cc5838f10f7c50e3ab32023-05-09T07:53:07ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752017-11-0120177511610.14232/ejqtde.2017.1.755912Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theoryRabil Ayazoglu (Mashiyev)0Ismail Ekincioglu1Gulizar Alisoy2Bayburt University, TurkeyFaculty of Science and Arts, Dumlupinar University, Kutahya, TurkeyFaculty of Science and Arts, Namik Kemal University, Tekirdag, TurkeyIn this paper, we deal with the following $p(x)$-Schrödinger problem: \begin{equation*} \begin{cases} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+V(x)\left\vert u\right\vert ^{p(x)-2}u=f(x,u) & \hbox{in $\mathbb{R}^{N}$ ;} \\ u\in W^{1,p(x)}(\mathbb{R}^{N}), & \hbox{} \end{cases} \end{equation*} where the nonlinearity is sublinear. We present the existence of infinitely many solutions for the problem. The main tool used here is a variational method and Krasnoselskii's genus theory combined with the theory of variable exponent Sobolev spaces. We also establish a Bartsch–Wang type compact embedding theorem for the variable exponent spaces.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5912$p(x)$-laplace operatorschrödinger equationvariable exponent lebesgue–sobolev spaceskrasnoselskii's genus |
| spellingShingle | Rabil Ayazoglu (Mashiyev) Ismail Ekincioglu Gulizar Alisoy Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory Electronic Journal of Qualitative Theory of Differential Equations $p(x)$-laplace operator schrödinger equation variable exponent lebesgue–sobolev spaces krasnoselskii's genus |
| title | Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory |
| title_full | Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory |
| title_fullStr | Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory |
| title_full_unstemmed | Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory |
| title_short | Multiple small solutions for $p(x)$-Schrödinger equations with local sublinear nonlinearities via genus theory |
| title_sort | multiple small solutions for p x schrodinger equations with local sublinear nonlinearities via genus theory |
| topic | $p(x)$-laplace operator schrödinger equation variable exponent lebesgue–sobolev spaces krasnoselskii's genus |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5912 |
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