The existence and multiplicity of weak solutions for some nonuniformly nonlinear elliptic systems are obtained by using the minimum principle and the Mountain-pass theorem.
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References
G. A. Afrouzi and Z. Naghizadeh, “An existence theorem for a class of nonuniformly nonlinear systems,” Austral. J. Basic & Appl. Sci., 5, No. 7, 1313–1317 (2011).
A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical points theory and applications,” J. Funct. Anal., 4, 349–381 (1973).
H. Brezis, Analyse Fonctionnelle: Theorie et Applications, Masson, Paris (1992).
D. G. Costa and C. A. Magalh aes, “Existence results for perturbations of the p-Laplacian,” Nonlin. Anal., 24, 409–418 (1995).
G. M. Figueiredo, “Existence of positive solutions for a class of p&q elliptic problems with critical growth on ℝN,” J. Math. Anal. Appl., 378, 507–518 (2011).
C. He and G. Li, “The existence of a nontrivial solution to the p&q -Laplacian problem with nonlinearity asymptotic to u p−1 at infinity in ℝN,” Nonlin. Anal., 68, 1100–1119 (2008).
O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, Acad. Press, New York (1973).
G. Li and X. Liang, “The existence of nontrivial solutions to nonlinear elliptic equation of p − q -Laplacian type on ℝN,” Nonlin. Anal., 71, 2316–2334 (2009).
G. Li and Z. Guo, “Multiple solutions for the p&q -Laplacian problem with critical exponent,” Acta Math. Sci., 29, 903–918 (2009).
Z. X. Li and Y. T. Shen, “Existence of nontrivial solutions for p-Laplacian-like equations,” Acta Math. Appl. Sinica., 27, No. 3, 393–406 (2011).
W. M. Ni and J. Serrin, “Nonexistence theorems for quasilinear partial differential equations,” Rend. Circ. Mat. Palermo (Suppl.), 8, 171–185 (1985).
W. M. Ni and J. Serrin, “Existence and nonexistence theorems for ground states for quasilinear partial differential equations,” Atti Conveg. Lincei, 77, 231–257 (1985).
J. M. Bezerra do Ó, “Existence of solutions for quasilinear elliptic equations,” J. Math. Anal. Appl., 207, 104–126 (1997).
M. Struwe, Variational Methods, 2nd ed., Springer-Verlag (2008).
M. Wu and Z. Yang, “A class of p − q -Laplacian type equation with potentials eigenvalue problem in ℝN,” Boundary Value Probl., 2009, ID 185319 (2009).
J. Zhang and Z. Zhang, “Existence results for some nonlinear elliptic systems,” Nonlin. Anal., 71, 2840–2846 (2009).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 9, pp. 1155–1165, September, 2014.
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Afrouzi, G.A., Naghizadeh, Z. & Chung, N.T. On a Class of Nonuniformly Nonlinear Systems with Dirichlet Boundary Conditions. Ukr Math J 66, 1289–1301 (2015). https://doi.org/10.1007/s11253-015-1009-0
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DOI: https://doi.org/10.1007/s11253-015-1009-0