Existence of solution for a class of biharmonic equations
In this paper, We prove the solvability of the biharmonic problem $$\begin{cases}\Delta^{2}u=f(x,u)+h ~~~~in~~\Omega, &\hbox{}\\ u=\Delta u=0 ~~~~~~on ~~\partial\Omega,\\\end{cases}$$ for a given function $h\in L^2(\Omega)$, if the limits at infinity of the quotients $f(x,s)/s$ and $2F(x,s)/s...
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Sociedade Brasileira de Matemática
2014-01-01
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Online Access: | http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/16178 |
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doaj-e91ab5af60d5414a81de45f578a9d6682020-11-24T23:49:20ZengSociedade Brasileira de MatemáticaBoletim da Sociedade Paranaense de Matemática0037-87122175-11882014-01-013219910810.5269/bspm.v32i1.161789303Existence of solution for a class of biharmonic equationsNajib Tsouli0Omar Chakrone1Omar Darhouche2Mostafa Rahmani3University Mohamed 1 Department of MathematicsUniversity Mohamed 1 Department of MathematicsUniversity Mohamed 1 Department of MathematicsUniversity Mohamed 1 Department of MathematicsIn this paper, We prove the solvability of the biharmonic problem $$\begin{cases}\Delta^{2}u=f(x,u)+h ~~~~in~~\Omega, &\hbox{}\\ u=\Delta u=0 ~~~~~~on ~~\partial\Omega,\\\end{cases}$$ for a given function $h\in L^2(\Omega)$, if the limits at infinity of the quotients $f(x,s)/s$ and $2F(x,s)/s$ for a.e.$x\in\Omega$ lie between two consecutive eigenvalues of the biharmonic operator $\Delta^2$, where $F(x,s)$ denotes the primitive $F(x,s)=\int_{0}^{s}{f(x,t)dt}$.http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/16178Biharmonic equationfourth elliptic equationnonresonance |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Najib Tsouli Omar Chakrone Omar Darhouche Mostafa Rahmani |
spellingShingle |
Najib Tsouli Omar Chakrone Omar Darhouche Mostafa Rahmani Existence of solution for a class of biharmonic equations Boletim da Sociedade Paranaense de Matemática Biharmonic equation fourth elliptic equation nonresonance |
author_facet |
Najib Tsouli Omar Chakrone Omar Darhouche Mostafa Rahmani |
author_sort |
Najib Tsouli |
title |
Existence of solution for a class of biharmonic equations |
title_short |
Existence of solution for a class of biharmonic equations |
title_full |
Existence of solution for a class of biharmonic equations |
title_fullStr |
Existence of solution for a class of biharmonic equations |
title_full_unstemmed |
Existence of solution for a class of biharmonic equations |
title_sort |
existence of solution for a class of biharmonic equations |
publisher |
Sociedade Brasileira de Matemática |
series |
Boletim da Sociedade Paranaense de Matemática |
issn |
0037-8712 2175-1188 |
publishDate |
2014-01-01 |
description |
In this paper, We prove the solvability of the biharmonic problem
$$\begin{cases}\Delta^{2}u=f(x,u)+h ~~~~in~~\Omega, &\hbox{}\\
u=\Delta u=0 ~~~~~~on ~~\partial\Omega,\\\end{cases}$$
for a given function $h\in L^2(\Omega)$, if the limits at infinity of the quotients $f(x,s)/s$ and $2F(x,s)/s$ for a.e.$x\in\Omega$ lie between two consecutive eigenvalues of the biharmonic operator $\Delta^2$, where $F(x,s)$ denotes the primitive $F(x,s)=\int_{0}^{s}{f(x,t)dt}$. |
topic |
Biharmonic equation fourth elliptic equation nonresonance |
url |
http://periodicos.uem.br/ojs/index.php/BSocParanMat/article/view/16178 |
work_keys_str_mv |
AT najibtsouli existenceofsolutionforaclassofbiharmonicequations AT omarchakrone existenceofsolutionforaclassofbiharmonicequations AT omardarhouche existenceofsolutionforaclassofbiharmonicequations AT mostafarahmani existenceofsolutionforaclassofbiharmonicequations |
_version_ |
1725482739487997952 |