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Solutions of elliptic problems of \(p\)-Laplacian type in a cylindrical symmetric domain. (English) Zbl 1299.35133

Summary: We consider the \(p\)-Laplacian type elliptic problem \[ \begin{cases} -\text{div} (a(x,\nabla u)) = h(x)| u| ^{q-2} u + g(x) & \text{in }\;\Omega,\\ u = 0 & \text{in }\;\partial\Omega, \end{cases} \] where \(\Omega = \Omega_1 \times \Omega_2 \subset \mathbb R^N\) is a bounded domain having cylindrical symmetry, \(\Omega_1 \subset \mathbb R^m\) is a bounded regular domain and \(\Omega_2\) is a \(k\)-dimensional ball of radius \(R\) centered in the origin, and \(m + k = N\), \(m \geqq 1\), \(k \geqq 2\). Under some suitable conditions on the functions \(a\) and \(h\), using variational methods we prove that the problem has at least one resp. at least two solutions in two cases: \(g = 0\) and \(g\neq 0\).

MSC:

35J62 Quasilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
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[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 4 (1973), 349–381. · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2] D. Arcoya and L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal., 134 (1996), 249–274. · Zbl 0884.58023 · doi:10.1007/BF00379536
[3] M. Badiale and E. Serra, Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Stud., 4 (2004), 453–467. · Zbl 1113.35047
[4] G. Bonanno, Some remarks on a three critical points theorem, Nonlinear Anal., 54 (2003), 651–665. · Zbl 1031.49006 · doi:10.1016/S0362-546X(03)00092-0
[5] A. Cano and M. Clapp, Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity, J. Diff. Eqns., 237 (2007), 133–158. · Zbl 1131.35029 · doi:10.1016/j.jde.2007.03.002
[6] A. Castro and M. Clapp, The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain, Nonlinearity, 16 (2003), 579–590. · Zbl 1108.35054 · doi:10.1088/0951-7715/16/2/313
[7] N. T. Chung and Q. A. Ngo, A multiplicity result for a class of equations of p-Laplacian type with sign-changing nonlinearities, Glasgow Math. J., 51 (2009), 513–524. · Zbl 1274.35131 · doi:10.1017/S001708950900514X
[8] P. De Nápoli and M. C. Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal., 54 (2003), 1205–1219. · Zbl 1274.35114 · doi:10.1016/S0362-546X(03)00105-6
[9] D. M. Duc, Nonlinear singular elliptic equations, J. London. Math. Soc., 40 (1989), 420–440. · Zbl 0661.35030 · doi:10.1112/jlms/s2-40.3.420
[10] D. M. Duc and N. T. Vu, Nonuniformly elliptic equations of p-Laplacian type, Nonlinear Anal., 61 (2005), 1483–1495. · Zbl 1125.35344 · doi:10.1016/j.na.2005.02.049
[11] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324–353. · Zbl 0286.49015 · doi:10.1016/0022-247X(74)90025-0
[12] A. Kristály, H. Lisei and C. Varga, Multiple solutions for p-Laplacian type equations, Nonlinear Anal., 68 (2008), 1375–1381. · Zbl 1136.35034 · doi:10.1016/j.na.2006.12.031
[13] M. Mihăilescu, Existence and multiplicity of weak solutions for a class of denegerate nonlinear elliptic equations, Boundary Value Problems, 2006, Art. ID 41295, 1–17.
[14] Q. A. Ngo, Existence results for a class of nonuniformly elliptic equations of p-Laplacian type, Anal. Appl., 7 (2009), 185–197. · Zbl 1175.35054 · doi:10.1142/S0219530509001323
[15] Q. A. Ngo and H. Q. Toan, Existence of solutions for a resonant problem under Landesman–Lazer conditions, Electron. J. Diff. Equ., 98 (2008), 1–10. · Zbl 1177.35078
[16] Q. A. Ngo and H. Q. Toan, Some remarks on a class of nonuniformly elliptic equations of p-Laplacian type, Acta Appl. Math., 106 (2009), 229–239. · Zbl 1170.35412 · doi:10.1007/s10440-008-9291-6
[17] E. Serra, Nonradial positive solutions for the Hénon equation with critical growth, Calc. Var., 23 (2005), 301–326. · Zbl 1207.35147 · doi:10.1007/s00526-004-0302-9
[18] D. Smets, J. Su and M. Willem, Nonradial ground states for the Hénon equation, Comm. Contemp. Math., 4 (2002), 467–480. · Zbl 1160.35415 · doi:10.1142/S0219199702000725
[19] H. Q. Toan and Q. A. Ngo, Multiplicity of weak solutions for a class of nonuniormly ellitptic equations of p-Laplacian type, Nonlinear Anal., 70 (2009), 1536–1546. · Zbl 1170.35415 · doi:10.1016/j.na.2008.02.033
[20] W. Wang, Sobolev embeddings involving symmetry, Bull. Sci. Math., 130 (2006), 269–278. · Zbl 1115.46031 · doi:10.1016/j.bulsci.2005.05.004
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