Multiple solutions to a class of inclusion problem with the \(p(x)\)-Laplacian. (English) Zbl 1247.35209
Summary: We obtain the existence of at least two nontrivial solutions for a nonlinear elliptic problem involving \(p(x)\)-Laplacian type operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.
MSC:
| 35R70 | PDEs with multivalued right-hand sides |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J70 | Degenerate elliptic equations |
| 35P30 | Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs |
Keywords:
\(p(x)\)-Laplacian; locally Lipschitz function; variable exponent Sobolev space; nonlinear eigenvalue problem; multiple solutionsReferences:
| [1] | Clarke FH, Optimization and Nonsmooth Analysis (1993) |
| [2] | DOI: 10.1016/j.na.2007.10.053 · Zbl 1152.35456 · doi:10.1016/j.na.2007.10.053 |
| [3] | DOI: 10.1016/j.na.2007.06.023 · Zbl 1387.35639 · doi:10.1016/j.na.2007.06.023 |
| [4] | DOI: 10.1016/j.na.2003.11.011 · doi:10.1016/j.na.2003.11.011 |
| [5] | DOI: 10.1016/j.jmaa.2005.02.029 · Zbl 1082.49008 · doi:10.1016/j.jmaa.2005.02.029 |
| [6] | DOI: 10.1016/j.na.2005.03.101 · Zbl 1130.35073 · doi:10.1016/j.na.2005.03.101 |
| [7] | DOI: 10.1016/j.na.2009.04.019 · Zbl 1175.35160 · doi:10.1016/j.na.2009.04.019 |
| [8] | DOI: 10.1016/j.nonrwa.2009.11.014 · Zbl 1196.35236 · doi:10.1016/j.nonrwa.2009.11.014 |
| [9] | DOI: 10.1016/j.nonrwa.2008.10.019 · Zbl 1181.35119 · doi:10.1016/j.nonrwa.2008.10.019 |
| [10] | DOI: 10.1002/mana.200310157 · Zbl 1065.46024 · doi:10.1002/mana.200310157 |
| [11] | DOI: 10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T · Zbl 1030.46033 · doi:10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T |
| [12] | Fan XL, J. Gansu Educ. Coll. 12 (1) pp 1– (1998) |
| [13] | DOI: 10.1006/jmaa.2000.7266 · Zbl 0988.46025 · doi:10.1006/jmaa.2000.7266 |
| [14] | DOI: 10.1006/jmaa.2000.7617 · Zbl 1028.46041 · doi:10.1006/jmaa.2000.7617 |
| [15] | Kovacik O, Czech. Math. J. 41 pp 592– (1991) |
| [16] | DOI: 10.1016/S0362-546X(02)00150-5 · Zbl 1146.35353 · doi:10.1016/S0362-546X(02)00150-5 |
| [17] | Chang KC, Critical Point Theory and Applications (1996) |
| [18] | DOI: 10.1017/S1446788700002202 · doi:10.1017/S1446788700002202 |
| [19] | DOI: 10.1016/j.jmaa.2008.05.086 · Zbl 1163.35026 · doi:10.1016/j.jmaa.2008.05.086 |
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