# zbMATH — the first resource for mathematics

Infinitely many solutions of some nonlinear variational equations. (English) Zbl 1160.49007
Summary: The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem ${\bar J}(u)=\int _\Omega {\bar A} (x,u)|\nabla u|^p \,dx - \int_\Omega G(x,u) \,dx$ in the Banach space $${W^{1,p}_0(\Omega) \cap L^\infty(\Omega)}$$, being $$\Omega$$ a bounded domain in $${\mathbb {R}^N}$$. In order to use “classical” theorems, a suitable variant of condition $$(C)$$ is proved and $${W^{1,p}_0(\Omega)}$$ is decomposed according to a “good” sequence of finite dimensional subspaces.

##### MSC:
 49J40 Variational inequalities 35J65 Nonlinear boundary value problems for linear elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
Full Text:
##### References:
 [1] Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 [2] Arcoya D., Boccardo L.: Critical points for multiple integrals of the calculus of variations. Arch. Ration. Mech. Anal. 134, 249–274 (1996) · Zbl 0884.58023 · doi:10.1007/BF00379536 [3] Arcoya D., Boccardo L.: Some remarks on critical point theory for nondifferentiable functionals. NoDEA Nonlinear Differ. Equ. Appl. 6, 79–100 (1999) · Zbl 0923.35049 · doi:10.1007/s000300050066 [4] Arcoya D., Boccardo L., Orsina L.: Existence of critical points for some noncoercive functionals. Ann. Inst. H. Poincaré Anal. Non Linéaire 18, 437–457 (2001) · Zbl 1035.49007 · doi:10.1016/S0294-1449(01)00069-5 [5] Bartolo P., Benci V., Fortunato D.: Abstract critical point theorems and applications to some nonlinear problems with ”strong” resonance at infinity. Nonlinear Anal. TMA 7, 981–1012 (1983) · Zbl 0522.58012 · doi:10.1016/0362-546X(83)90115-3 [6] Boccardo L., Murat F., Puel J.P.: Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. IV Ser. 152, 183–196 (1988) · Zbl 0687.35042 · doi:10.1007/BF01766148 [7] Browder F.E.: Existence theorems for nonlinear partial differential equations. In: Chern, S.S., Smale, S.(eds) Proceedings of Symposia in Pure Mathematics, vol. XVI, pp. 1–60. AMS, Providence (1970) · Zbl 0211.17204 [8] Candela A.M., Palmieri G.: Multiple solutions of some nonlinear variational problems. Adv. Nonlinear Stud. 6, 269–286 (2006) · Zbl 1102.49006 [9] Candela, A.M., Palmieri, G.: Some abstract critical point theorems and applications (preprint) · Zbl 1189.58003 [10] Canino A.: Multiplicity of solutions for quasilinear elliptic equations. Topol. Methods Nonlinear Anal. 6, 357–370 (1995) · Zbl 0863.35038 [11] Cerami G.: Un criterio di esistenza per i punti critici su varietà illimitate. Istit. Lombardo Accad. Sci. Lett. Rend. A 112, 332–336 (1978) · Zbl 0436.58006 [12] Dacorogna B.: Direct Methods in the Calculus of Variations. Springer, New York (1989) · Zbl 0703.49001 [13] Degiovanni M., Lancelotti S.: Linking over cones and nontrivial solutions for p–Laplace equations with p–superlinear nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 907–919 (2007) · Zbl 1132.35040 · doi:10.1016/j.anihpc.2006.06.007 [14] Dinca G., Jebelean P., Mawhin J.: Variational and topological methods for Dirichlet problems with p–Laplacian. Port. Math. (N.S.) 58, 339–378 (2001) · Zbl 0991.35023 [15] Ladyzhenskaya O.A., Ural’tseva N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968) · Zbl 0164.13002 [16] Lindqvist P.: On the equation $${{\mathrm div} (|\nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2}u =0}$$ . Proc. Am. Math. Soc. 109, 157–164 (1990) · Zbl 0714.35029 [17] Marzocchi M.: Multiple solutions of quasilinear equations involving an area–type term. J. Math. Anal. Appl. 196, 1093–1104 (1995) · Zbl 0854.35042 · doi:10.1006/jmaa.1995.1462 [18] Palais R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963) · Zbl 0122.10702 · doi:10.1016/0040-9383(63)90013-2 [19] Pellacci B., Squassina M.: Unbounded critical points for a class of lower semicontinuous functionals. J. Diff. Equ. 201, 25–62 (2004) · Zbl 1330.35103 · doi:10.1016/j.jde.2004.03.002 [20] Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Reg. Conf. Ser. Math., vol. 65. AMS, Providence (1986) · Zbl 0609.58002 [21] Struwe M.: Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, 3rd edn. Springer, Berlin (1996) · Zbl 0864.49001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.