The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities
In this paper, we study a nonlinear boundary value system with $p$-Laplacian operator $$\left\{\begin{array}{lll} (\phi_{p_1}(u'))'+a_1(t)f(u,v)=0, 0<t<1,\cr (\phi_{p_2}(v'))'+a_2(t)g(u,v)=0, 0<t<1,\cr \alpha_1\phi_{p_1}(u(0))-\beta_1\phi_{p_1}(u'(0))=\gamma_1\p...
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| Format: | Article |
| Language: | English |
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University of Szeged
2010-12-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=539 |
| _version_ | 1797830708504297472 |
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| author | Fuyi Xu |
| author_facet | Fuyi Xu |
| author_sort | Fuyi Xu |
| collection | DOAJ |
| description | In this paper, we study a nonlinear boundary value system with $p$-Laplacian operator
$$\left\{\begin{array}{lll}
(\phi_{p_1}(u'))'+a_1(t)f(u,v)=0, 0<t<1,\cr
(\phi_{p_2}(v'))'+a_2(t)g(u,v)=0, 0<t<1,\cr
\alpha_1\phi_{p_1}(u(0))-\beta_1\phi_{p_1}(u'(0))=\gamma_1\phi_{p_1}(u(1))+\delta_1\phi_{p_1}(u'(1))=0,
\alpha_2\phi_{p_2}(v(0))-\beta_2\phi_{p_2}(v'(0))=\gamma_2\phi_{p_2}(v(1))+\delta_2\phi_{p_2}(v'(1))=0,
\end{array}\right.$$
where $\phi_{p_i}(s)=|s|^{p_i-2}s, p_i>1, i=1,2$. We obtain some sufficient conditions for the existence of two positive solutions or infinitely many positive solutions by using a fixed-point theorem in cones. Especially, the nonlinear terms $f,g $ are allowed to change sign. The conclusions essentially extend and improve the known results. |
| first_indexed | 2024-04-09T13:40:24Z |
| format | Article |
| id | doaj.art-50e30e4cee6346299444d39203b2fcf9 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:40:24Z |
| publishDate | 2010-12-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-50e30e4cee6346299444d39203b2fcf92023-05-09T07:53:00ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752010-12-0120107911410.14232/ejqtde.2010.1.79539The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearitiesFuyi Xu0Shandong University of Technology, Zibo, Shandong, P. R. ChinaIn this paper, we study a nonlinear boundary value system with $p$-Laplacian operator $$\left\{\begin{array}{lll} (\phi_{p_1}(u'))'+a_1(t)f(u,v)=0, 0<t<1,\cr (\phi_{p_2}(v'))'+a_2(t)g(u,v)=0, 0<t<1,\cr \alpha_1\phi_{p_1}(u(0))-\beta_1\phi_{p_1}(u'(0))=\gamma_1\phi_{p_1}(u(1))+\delta_1\phi_{p_1}(u'(1))=0, \alpha_2\phi_{p_2}(v(0))-\beta_2\phi_{p_2}(v'(0))=\gamma_2\phi_{p_2}(v(1))+\delta_2\phi_{p_2}(v'(1))=0, \end{array}\right.$$ where $\phi_{p_i}(s)=|s|^{p_i-2}s, p_i>1, i=1,2$. We obtain some sufficient conditions for the existence of two positive solutions or infinitely many positive solutions by using a fixed-point theorem in cones. Especially, the nonlinear terms $f,g $ are allowed to change sign. The conclusions essentially extend and improve the known results.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=539 |
| spellingShingle | Fuyi Xu The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities Electronic Journal of Qualitative Theory of Differential Equations |
| title | The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities |
| title_full | The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities |
| title_fullStr | The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities |
| title_full_unstemmed | The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities |
| title_short | The existence of positive solutions for nonlinear boundary system with $p$-Laplacian operator based on sign-changing nonlinearities |
| title_sort | existence of positive solutions for nonlinear boundary system with p laplacian operator based on sign changing nonlinearities |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=539 |
| work_keys_str_mv | AT fuyixu theexistenceofpositivesolutionsfornonlinearboundarysystemwithplaplacianoperatorbasedonsignchangingnonlinearities AT fuyixu existenceofpositivesolutionsfornonlinearboundarysystemwithplaplacianoperatorbasedonsignchangingnonlinearities |