Linking over cones and nontrivial solutions for $p$-Laplace equations with $p$-superlinear nonlinearity

Degiovanni, Marco; Lancelotti, Sergio

Annales de l'I.H.P. Analyse non linéaire, Tome 24 (2007) no. 6, pp. 907-919.

[1] Ambrosetti A., Rabinowitz P.H., Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349-381. | MR | Zbl

[2] Anane A., Simplicité et isolation de la première valeur propre du p-laplacien avec poids, C. R. Acad. Sci. Paris Sér. I Math. 305 (16) (1987) 725-728. | MR | Zbl

[3] Anane A., Tsouli N., On the second eigenvalue of the p-Laplacian, in: Nonlinear Partial Differential Equations, Fès, 1994, Pitman Res. Notes Math. Ser., vol. 343, Longman, Harlow, 1996, pp. 1-9. | MR | Zbl

[4] Bonnet A., A deformation lemma on a $C^1$ manifold, Manuscripta Math. 81 (3-4) (1993) 339-359. | EuDML | MR | Zbl

[5] Canino A., Degiovanni M., Nonsmooth critical point theory and quasilinear elliptic equations, in: Topological Methods in Differential Equations and Inclusions, Montreal, PQ, 1994, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 472, Kluwer Acad. Publ., Dordrecht, 1995, pp. 1-50. | MR | Zbl

[6] Chang K.-C., Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, vol. 6, Birkhäuser Boston Inc., Boston, MA, 1993. | MR | Zbl

[7] Cingolani S., Degiovanni M., Nontrivial solutions for p-Laplace equations with right-hand side having p-linear growth at infinity, Comm. Partial Differential Equations 30 (8) (2005) 1191-1203. | MR | Zbl

[8] Corvellec J.-N., Degiovanni M., Marzocchi M., Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal. 1 (1) (1993) 151-171. | MR | Zbl

[9] Cuesta M., Eigenvalue problems for the p-Laplacian with indefinite weights, Electron. J. Differential Equations 33 (2001), 9 p. (electronic). | EuDML | MR | Zbl

[10] Degiovanni M., On Morse theory for continuous functionals, Conf. Semin. Mat. Univ. Bari (290) (2003) 1-22. | MR

[11] Del Pino M., Elgueta M., Manásevich R., A homotopic deformation along p of a Leray-Schauder degree result and existence for ${\left({\left|u^{\text{'}}\right|}^{p-2}{u}^{\text{'}}\right)}^{\text{'}}+f(t,u)=0$, $u\left(0\right)=u\left(T\right)=0$, $p>1$, J. Differential Equations 80 (1) (1989) 1-13. | Zbl

[12] Dinca G., Jebelean P., Mawhin J., Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. (N.S.) 58 (3) (2001) 339-378. | MR | Zbl

[13] Drábek P., Robinson S.B., Resonance problems for the p-Laplacian, J. Funct. Anal. 169 (1) (1999) 189-200. | MR | Zbl

[14] Fadell E.R., Rabinowitz P.H., Bifurcation for odd potential operators and an alternative topological index, J. Funct. Anal. 26 (1) (1977) 48-67. | MR | Zbl

[15] Fadell E.R., Rabinowitz P.H., Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (2) (1978) 139-174. | MR | Zbl

[16] Fan X., Li Z., Linking and existence results for perturbations of the p-Laplacian, Nonlinear Anal. 42 (8) (2000) 1413-1420. | MR | Zbl

[17] Frigon M., On a new notion of linking and application to elliptic problems at resonance, J. Differential Equations 153 (1) (1999) 96-120. | MR | Zbl

[18] García Azorero J., Peral Alonso I., Comportement asymptotique des valeurs propres du p-laplacien, C. R. Acad. Sci. Paris Sér. I Math. 307 (2) (1988) 75-78. | MR | Zbl

[19] Ioffe A., Schwartzman E., Metric critical point theory. I. Morse regularity and homotopic stability of a minimum, J. Math. Pures Appl. (9) 75 (2) (1996) 125-153. | MR | Zbl

[20] Lindqvist P., On the equation $\mathrm{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)+\lambda {\left|u\right|}^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1) (1990) 157-164. | MR | Zbl

[21] Lindqvist P., Addendum: “On the equation $\mathrm{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)+\lambda {\left|u\right|}^{p-2}u=0$”, Proc. Amer. Math. Soc. 116 (2) (1992) 583-584. | Zbl

[22] Liu S., Existence of solutions to a superlinear p-Laplacian equation, Electron. J. Differential Equations 66 (2001), 6 p. (electronic). | MR | Zbl

[23] Marino A., Micheletti A.M., Pistoia A., Some variational results on semilinear problems with asymptotically nonsymmetric behaviour, in: Nonlinear Analysis, Sc. Norm. Super. di Pisa Quaderni, Scuola Norm. Sup., Pisa, 1991, pp. 243-256. | MR | Zbl

[24] Marino A., Micheletti A.M., Pistoia A., A nonsymmetric asymptotically linear elliptic problem, Topol. Methods Nonlinear Anal. 4 (2) (1994) 289-339. | MR | Zbl

[25] Perera K., Nontrivial solutions of p-superlinear p-Laplacian problems, Appl. Anal. 82 (9) (2003) 883-888. | MR | Zbl

[26] Perera K., Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2) (2003) 301-309. | MR | Zbl

[27] Perera K., Szulkin A., p-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst. 13 (3) (2005) 743-753. | MR | Zbl

[28] Rabinowitz P.H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. | MR | Zbl

[29] Ribarska N.K., Tsachev T.Y., Krastanov M.I., Deformation lemma, Ljusternik-Schnirellmann theory and mountain pass theorem on $C^1$-Finsler manifolds, Serdica Math. J. 21 (3) (1995) 239-266. | Zbl

[30] Spanier E.H., Algebraic Topology, McGraw-Hill Book Co., New York, 1966. | MR | Zbl

[31] Szulkin A., Ljusternik-Schnirelmann theory on $C^1$-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (2) (1988) 119-139. | Numdam | Zbl

[32] Szulkin A., Willem M., Eigenvalue problems with indefinite weight, Studia Math. 135 (2) (1999) 191-201. | MR | Zbl