Existence and multiplicity of solutions for p-Laplacian Neumann problems. (English) Zbl 1419.35051
Summary: In this paper, existence theorems are proved for p-Laplacian Neumann problems under the Landesman-Lazer type condition. Our results are derived from a classical saddle point theorem and the least action principle respectively. Furthermore, multiplicity of solutions is established by applying a known multiple critical points result due to H. Brezis and L. Nirenberg. The above-mentioned conclusions are based on variational methods.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35J50 | Variational methods for elliptic systems |
| 49J35 | Existence of solutions for minimax problems |
Keywords:
\(p\)-Laplacian; Neumann problem; Landesman-Lazer type condition; saddle point theorem; least action principleReferences:
| [1] | Aizicovici, S., Papageorgiou, N.S., Staicu, V.: On a p-superlinear Neumann p-Laplacian equation. Topol. Methods Nonlinear Anal. 34, 111-130 (2009) · Zbl 1183.35099 · doi:10.12775/TMNA.2009.032 |
| [2] | Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Existence of multiple solutions with precise sign information for superlinear Neumann problems. Ann. Mat. Pura Appl. (4) 188, 679-719 (2009) · Zbl 1188.35057 · doi:10.1007/s10231-009-0096-7 |
| [3] | Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Multiple solutions for superlinear p-Laplacian Neumann problems. Osaka J. Math. 49, 699-740 (2012) · Zbl 1260.35037 |
| [4] | Anello, G.: Existence of infinitely many weak solutions for a Neumann problem. Nonlinear Anal. 57, 199-209 (2004) · Zbl 1058.35071 · doi:10.1016/j.na.2004.02.009 |
| [5] | Binding, P.A., Drábek, P., Huang, Y.X.: Existence of multiple solutions of critical quasilinear elliptic Neumann problems. Nonlinear Anal. Ser. A Theory Methods 42, 613-629 (2000) · Zbl 0964.35064 · doi:10.1016/S0362-546X(99)00118-2 |
| [6] | Bonanno, G., Candito, P.: Three solutions to a Neumann problem for elliptic equations involving the p-Laplacian. Arch. Math. (Basel) 80, 424-429 (2003) · Zbl 1161.35382 · doi:10.1007/s00013-003-0479-8 |
| [7] | Bonanno, G., Sciammetta, A.: Existence and multiplicity results to Neumann problems for elliptic equations involving the p-Laplacian. J. Math. Anal. Appl. 390, 59-67 (2012) · Zbl 1246.35080 · doi:10.1016/j.jmaa.2012.01.012 |
| [8] | Brezis, H., Nirenberg, L.: Remarks on finding critical points. Comm. Pure Appl. Math. 44, 939-963 (1991) · Zbl 0751.58006 · doi:10.1002/cpa.3160440808 |
| [9] | Cammaroto, F., Chinnì, A., Bella, B.D.: Some multiplicity results for quasilinear Neumann problems. Arch. Math. (Basel) 86, 154-162 (2006) · Zbl 1206.35127 · doi:10.1007/s00013-005-1425-8 |
| [10] | Filippakis, M., Gasiński, L., Papageorgiou, N.S.: Multiplicity results for nonlinear Neumann problems. Can. J. Math. 58, 64-92 (2006) · Zbl 1209.35161 · doi:10.4153/CJM-2006-004-6 |
| [11] | He, T., Chen, C., Huang, Y., Hou, C.: Infinitely many sign-changing solutions for p-Laplacian Neumann problems with indefinite weight. Appl. Math. Lett. 39, 73-79 (2015) · Zbl 1317.35097 · doi:10.1016/j.aml.2014.08.011 |
| [12] | Iannacci, R., Nkashama, M.N.: Nonlinear two point boundary value problem at resonance without Landesman-Lazer condition. Proc. Am. Math. Soc. 10, 943-952 (1989) · Zbl 0684.34025 |
| [13] | Liao, K., Tang, C.L.: Existence and multiplicity of periodic solutions for the ordinary p-Laplacian systems. J. Appl. Math. Comput. 35, 395-406 (2011) · Zbl 1218.34048 · doi:10.1007/s12190-009-0364-0 |
| [14] | Motreanu, D., Papageorgiou, N.S.: Existence and multiplicity of solutions for Neumann problems. J. Differ. Equ. 232, 1-35 (2007) · Zbl 1387.35159 · doi:10.1016/j.jde.2006.09.008 |
| [15] | Papageorgiou, N.S., Radulescu, V.D.: Multiplicity of solutions for resonant neumann problems with an indefinite and unbounded potential. Trans. Am. Math. Soc. 367(12), 8723-8756 (2015) · Zbl 1341.35028 · doi:10.1090/S0002-9947-2014-06518-5 |
| [16] | Papageorgiou, N.S., Radulescu, V.D.: Neumann problems with indefinite and unbounded potential and concave terms. Proc. Am. Math. Soc. 143(11), 4803-4816 (2015) · Zbl 1331.35116 · doi:10.1090/proc/12600 |
| [17] | Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. In: CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence (1986) · Zbl 0609.58002 |
| [18] | Ricceri, B.: Infinitely many solutions of the Neumann problem for elliptic equations involving the p-Laplacian. Bull. Lond. Math. Soc. 33, 331-340 (2001) · Zbl 1035.35031 · doi:10.1017/S0024609301008001 |
| [19] | Tang, C.L.: Solvability of Neumann problem for elliptic equation at resonance. Nonlinear Anal. 44, 323-335 (2001) · Zbl 1002.35047 · doi:10.1016/S0362-546X(99)00266-7 |
| [20] | Tang, C.L.: Some existence theorems for sublinear Neumann boundary value problem. Nonlinear Anal. 48, 1003-1011 (2002) · Zbl 1033.35029 · doi:10.1016/S0362-546X(00)00230-3 |
| [21] | Wu, X., Tan, K.K.: On existence and multiplicity of solutions of Neumann boundary value problems for quasi-linear elliptic equations. Nonlinear Anal. 65(7), 1334-1347 (2006) · Zbl 1109.35047 · doi:10.1016/j.na.2005.10.010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
