Existence and multiplicity of solutions for anisotropic elliptic equation
DOI:
https://doi.org/10.5269/bspm.45963Abstract
In this article we study the nonlinear problem
$$\left\{ \begin{array}{lr}
-\sum_{i=1}^{N}\partial_{x_{i}}a_{i}(x,\partial_{x_{i}}u)+ b(x)~|u|^{P_{+}^{+}-2}u =\lambda f(x,u) \quad in \quad \Omega\\
u=0 \qquad on \qquad \partial\Omega
\end{array} \right.$$
Using the variational method, under appropriate assumptions on $f$, we obtain a result on existence and multiplicity of solutions.
References
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15. B. Kone, S. Ouaro, and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Differ. Equ. 2009 (2009), 1-11.
16. M. Mihailescu; Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425. https://doi.org/10.1016/j.na.2006.07.027
17. M. Mihailescu, G. Moro¸sanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applicable Analysis 89 (2010), 257-271. https://doi.org/10.1080/00036810802713826
18. M. Mihailescu, P. Pucci, V. Rˇadulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687-698. https://doi.org/10.1016/j.jmaa.2007.09.015
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20. B. Ricceri, On three critical points theorem, Arch. Math. (Basel) 75 (2000), 220-226. https://doi.org/10.1007/s000130050496
21. X. Shi, X. Ding; Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem, Nonlinear Anal., 70 (2009), 3715-3720. https://doi.org/10.1016/j.na.2008.07.027
22. M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1
2. A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal. 14 (1973), 349-381. https://doi.org/10.1016/0022-1236(73)90051-7
3. G. Bonanno, A minimax inequality and its applications to ordinary differential equations. J. Math. Anal. Appli. 270(2002) 210-219. https://doi.org/10.1016/S0022-247X(02)00068-9
4. G. Bonanno, P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian, Arch. Math. (Basel) 80 (2003) 424-429. https://doi.org/10.1007/s00013-003-0479-8
5. M. M. Boureanu, Infinitely many solutions for a class of degenerate anisotropic elliptic problems with variable exponent, Taiwanese Journal of Mathematics 15 (2011), 2291-2310. https://doi.org/10.11650/twjm/1500406435
6. D. E. Edmunds, J. Rakosnık, Sobolev embedding with variable exponent, Studia Math. 143 (2000), 267-293. https://doi.org/10.4064/sm-143-3-267-293
7. A. R. El Amrouss, F. Mordi, and M. Moussaoui, Existence of solutions for fourth-order PDEs with variable exponents, Electron. J. Differ. Equ. 2009 (2009), No. 153. pp. 1-13.
8. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0
9. X. L. Fan, Anisotropic variable exponent Sobolev spaces and −→p (x)−Laplacian equations, Complex Var. Elliptic Equ. 56 (7-9) (2011), 623-642. https://doi.org/10.1080/17476931003728412
10. X. L. Fan, X. Y. Han, Existence and multiplicity of solutions for p(x) − Laplacian equations in RN , Nonlinear Anal. 59 (2004), 173-188. https://doi.org/10.1016/S0362-546X(04)00254-8
11. X. L. Fan, J. S. Shen, D. Zhao, Sobolev embedding theorems for spaces Wk,p(x) , J. Math. Anal. Appl. 262 (2001), 749-760. https://doi.org/10.1006/jmaa.2001.7618
12. X. L. Fan, D. Zhao, On the spaces L p(x) and W m,p(x) , J. Math. Anal. Appl. 263 (2001), 424-446. https://doi.org/10.1006/jmaa.2000.7617
13. Q. Liu; Existence of three solutions for p(x)-Laplacian equations, Nonlinear Anal., 68 (2008), pp. 2119-2127. https://doi.org/10.1016/j.na.2007.01.035
14. C. Ji, Remarks on the existence of three solutions for the p(x) − Laplacian equations, Nonlinear Anal.74 (2011), 2908-2915. https://doi.org/10.1016/j.na.2010.12.013
15. B. Kone, S. Ouaro, and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Differ. Equ. 2009 (2009), 1-11.
16. M. Mihailescu; Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator, Nonlinear Anal., 67 (2007), 1419-1425. https://doi.org/10.1016/j.na.2006.07.027
17. M. Mihailescu, G. Moro¸sanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applicable Analysis 89 (2010), 257-271. https://doi.org/10.1080/00036810802713826
18. M. Mihailescu, P. Pucci, V. Rˇadulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687-698. https://doi.org/10.1016/j.jmaa.2007.09.015
19. B. Ricceri, A three critical points theorem revisited. Nonlinear Anal. 70 (2009) 3084-3089. https://doi.org/10.1016/j.na.2008.04.010
20. B. Ricceri, On three critical points theorem, Arch. Math. (Basel) 75 (2000), 220-226. https://doi.org/10.1007/s000130050496
21. X. Shi, X. Ding; Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem, Nonlinear Anal., 70 (2009), 3715-3720. https://doi.org/10.1016/j.na.2008.07.027
22. M. Willem, Minimax Theorems, Birkhauser, Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1
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2021-12-20
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