Carl, S.; Dietrich, H. The weak upper and lower solution method for quasilinear elliptic equations with generalized subdifferentiable perturbations. (English) Zbl 0832.35039 Appl. Anal. 56, No. 3-4, 263-278 (1995). A variational approach to the method of upper and lower solution is suggested which allows to treat nonlinear elliptic boundary value problems with Baire-measurable lower order nonlinearities. To this end an associated multivalued setting of the problem is considered. First, we prove the existence of solutions of a ‘truncated’ auxiliary problem which is related to the minimization of a nonsmooth functional whose critical points are shown to be solutions of this auxiliary problem. Then it is shown that any solution of the auxiliary problem solves the original one. The existence of critical points of the functional under consideration is proved by showing that it satisfies a generalized Palais-Smale condition which is suggested by the variational principle of Ekeland. Reviewer: S.Carl (Halle) Cited in 21 Documents MSC: 35J60 Nonlinear elliptic equations 35R05 PDEs with low regular coefficients and/or low regular data 49K24 Optimal control problems with differential inclusions (nec./ suff.) (MSC2000) 35R70 PDEs with multivalued right-hand sides 35J25 Boundary value problems for second-order elliptic equations Keywords:generalized subdifferential; generalized Palais-Smale condition; multifunctions; measurable lower order nonlinearities PDFBibTeX XMLCite \textit{S. Carl} and \textit{H. Dietrich}, Appl. Anal. 56, No. 3--4, 263--278 (1995; Zbl 0832.35039) Full Text: DOI References: [1] Appell J., Nonlinear superposition operators (1990) · Zbl 0701.47041 [2] DOI: 10.1002/mana.19881380104 · Zbl 0666.35032 [3] DOI: 10.1016/0362-546X(92)90014-6 · Zbl 0755.35039 [4] Carl S., Differential Integral Equations 5 pp 581– (1992) [5] DOI: 10.1016/0022-247X(81)90095-0 · Zbl 0487.49027 [6] DOI: 10.1016/0022-0396(83)90018-9 · Zbl 0533.35088 [7] DOI: 10.1002/cpa.3160330203 · Zbl 0405.35074 [8] Clarke F.H., Optimization and Nonsmooth Analysis (1983) · Zbl 0582.49001 [9] DOI: 10.1016/0022-247X(90)90226-6 · Zbl 0717.49007 [10] Deuel J., Proc. Royal Soc. Edinburgh 74 pp 49– (1974) [11] DOI: 10.1016/0022-247X(74)90025-0 · Zbl 0286.49015 [12] DOI: 10.1090/S0273-0979-1979-14595-6 · Zbl 0441.49011 [13] DOI: 10.1093/imamat/44.2.181 · Zbl 0705.35047 [14] Okochi H., Hiroshima Math. J. 22 pp 237– (1992) [15] Pascali D., Nonlinear mappings of monotone type (1978) · Zbl 0423.47021 [16] DOI: 10.1007/BF01174897 · Zbl 0403.35036 [17] DOI: 10.1112/jlms/s2-21.2.319 · Zbl 0434.35042 [18] DOI: 10.1112/jlms/s2-21.2.329 · Zbl 0434.35043 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.