Existence of infinitely many weak solutions for some quasilinear $\vec {p}(x)$-elliptic Neumann problems

Ahmed Ahmed; Taghi Ahmedatt; Hassane Hjiaj; Abdelfattah Touzani

Mathematica Bohemica (2017)

  • Volume: 142, Issue: 3, page 243-262
  • ISSN: 0862-7959

Abstract

top
We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) &\text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 &\text {on} \ \partial \Omega . \end {cases} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.

How to cite

top

Ahmed, Ahmed, et al. "Existence of infinitely many weak solutions for some quasilinear $\vec {p}(x)$-elliptic Neumann problems." Mathematica Bohemica 142.3 (2017): 243-262. <http://eudml.org/doc/294102>.

@article{Ahmed2017,
abstract = {We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin \{cases\} -\displaystyle \sum \_\{i=1\}^\{N\}\frac \{\partial \}\{\partial x\_\{i\}\}a\_\{i\}\Big (x,\frac \{\partial u\}\{\partial x\_\{i\}\}\Big ) + b(x)|u|^\{p\_\{0\}(x)-2\}u = f(x,u)+ g(x,u) &\text \{in\} \ \Omega , \\ \quad \dfrac \{\partial u\}\{\partial \gamma \} = 0 &\text \{on\} \ \partial \Omega . \end \{cases\} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.},
author = {Ahmed, Ahmed, Ahmedatt, Taghi, Hjiaj, Hassane, Touzani, Abdelfattah},
journal = {Mathematica Bohemica},
language = {eng},
number = {3},
pages = {243-262},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of infinitely many weak solutions for some quasilinear $\vec \{p\}(x)$-elliptic Neumann problems},
url = {http://eudml.org/doc/294102},
volume = {142},
year = {2017},
}

TY - JOUR
AU - Ahmed, Ahmed
AU - Ahmedatt, Taghi
AU - Hjiaj, Hassane
AU - Touzani, Abdelfattah
TI - Existence of infinitely many weak solutions for some quasilinear $\vec {p}(x)$-elliptic Neumann problems
JO - Mathematica Bohemica
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 142
IS - 3
SP - 243
EP - 262
AB - We consider the following quasilinear Neumann boundary-value problem of the type $$ \begin {cases} -\displaystyle \sum _{i=1}^{N}\frac {\partial }{\partial x_{i}}a_{i}\Big (x,\frac {\partial u}{\partial x_{i}}\Big ) + b(x)|u|^{p_{0}(x)-2}u = f(x,u)+ g(x,u) &\text {in} \ \Omega , \\ \quad \dfrac {\partial u}{\partial \gamma } = 0 &\text {on} \ \partial \Omega . \end {cases} $$ We prove the existence of infinitely many weak solutions for our equation in the anisotropic variable exponent Sobolev spaces and we give some examples.
LA - eng
UR - http://eudml.org/doc/294102
ER -

References

top
  1. Anello, G., Cordaro, G., 10.1007/s00013-002-8314-1, Arch. Math. 79 (2002), 274-287. (2002) Zbl1091.35025MR1944952DOI10.1007/s00013-002-8314-1
  2. Azroul, E., Barbara, A., Benboubker, M. B., Hjiaj, H., 10.4064/am41-2-3, Appl. Math. 41 (2014), 149-163. (2014) Zbl1316.35107MR3281367DOI10.4064/am41-2-3
  3. Bendahmane, M., Chrif, M., Manouni, S. El, 10.4171/ZAA/1438, Z. Anal. Anwend. 30 (2011), 341-353. (2011) Zbl1231.35065MR2819499DOI10.4171/ZAA/1438
  4. Boureanu, M.-M., Rădulescu, V. D., 10.1016/j.na.2011.09.033, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 4471-4482. (2012) Zbl1262.35090MR2927115DOI10.1016/j.na.2011.09.033
  5. Nardo, R. Di, Feo, F., Guibé, O., Uniqueness result for nonlinear anisotropic elliptic equations, Adv. Differ. Equ. 18 (2013), 433-458. (2013) Zbl1272.35092MR3086461
  6. Diening, L., Harjulehto, P., Hästö, P., Růžička, M., 10.1007/978-3-642-18363-8, Lecture Notes in Mathematics 2017. Springer, Berlin (2011). (2011) Zbl1222.46002MR2790542DOI10.1007/978-3-642-18363-8
  7. Fan, X., Ji, C., 10.1016/j.jmaa.2006.12.055, J. Math. Anal. Appl. 334 (2007), 248-260. (2007) Zbl1157.35040MR2332553DOI10.1016/j.jmaa.2006.12.055
  8. Guibé, O., Uniqueness of the renormalized solution to a class of nonlinear elliptic equations, On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments (A. Alvino et al., eds.) Quaderni di Matematica 23. Caserta (2008), 255-282. (2008) Zbl1216.35036MR2762168
  9. Harjulehto, P., Hästö, P., 10.5565/PUBLMAT_52208_05, Publ. Mat., Barc. 52 (2008), 347-363. (2008) Zbl1163.46022MR2436729DOI10.5565/PUBLMAT_52208_05
  10. Kone, B., Ouaro, S., Traore, S., Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Differ. Equ. (electronic only) 2009 (2009), paper No. 144, 11 pages. (2009) Zbl1182.35092MR2565886
  11. Kováčik, O., Rákosník, J., On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czech. Math. J. 41 (1991), 592-618. (1991) Zbl0784.46029MR1134951
  12. Mihăilescu, M., Moroşanu, G., 10.1080/00036810802713826, Appl. Anal. 89 (2010), 257-271. (2010) Zbl1187.35074MR2598814DOI10.1080/00036810802713826
  13. Ricceri, B., 10.1016/S0377-0427(99)00269-1, J. Comput. Appl. Math. 113 (2000), 401-410. (2000) Zbl0946.49001MR1735837DOI10.1016/S0377-0427(99)00269-1
  14. užička, M. R\accent23, 10.1007/BFb0104029, Lecture Notes in Mathematics 1748. Springer, Berlin (2000). (2000) Zbl0962.76001MR1810360DOI10.1007/BFb0104029
  15. Zhao, L., Zhao, P., Xie, X., Existence and multiplicity of solutions for divergence type elliptic equations, Electron. J. Differ. Equ. (electronic only) 2011 (2011), paper No. 43, 9 pages. (2011) Zbl1213.35227MR2788662

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.