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 |
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.