Existence and uniqueness of solutions of nonlinear Neumann problems. (English) Zbl 0633.35042
Existence of non-trivial solutions is proved by means of upper and lower solutions. The solutions are then classified and uniqueness is obtained in certain classes. The results are optimal in the sense that our conditions are necessary and sufficient. We also deduce existence results for problems in \({\mathbb{R}}^ N\). Related papers are by the authors [Trans. Am. Math. Soc. 303, 487-501 (1987; Zbl 0633.35041)] and by J. Spruck [Commun. Partial Differ. Equations 8, 1605-1620 (1983; Zbl 0534.35055)].
MSC:
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 35B50 | Maximum principles in context of PDEs |
| 35B05 | Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs |
References:
| [1] | Amann, H.: On the existence of positive solutions of nonlinear elliptic boundary value problems. Indiana Univ. Math. J.21, 125-146 (1971) · Zbl 0219.35037 · doi:10.1512/iumj.1971.21.21012 |
| [2] | Aronson, D.G., Crandall, M., Peletier, L.A.: Stabilization of solutions of a degenerate diffusion problem. Nonlinear Anal. Theory Methods Appl.6, 1001-1022 (1982) · Zbl 0518.35050 · doi:10.1016/0362-546X(82)90072-4 |
| [3] | Bandle, C.: Isoperimetric inequalities and applications. Monographs and Studies in Mathematics7. Boston-London-Melbourne: Pitman 1980 · Zbl 0436.35063 |
| [4] | Bandle, C., Pozio, M.A., Tesei, A.: The asymptotic behaviour of the solutions of degenerate parabolic equations. Trans. Am. Math. Soc.303, 487-501 (1987) · Zbl 0633.35041 · doi:10.1090/S0002-9947-1987-0902780-3 |
| [5] | Bandle, C., Sperb, R.B., Stakgold, I.: Diffusion and reaction with monotone kynetics. Nonlinear Anal. TMA8, 321-333 (1984) · Zbl 0545.35011 · doi:10.1016/0362-546X(84)90034-8 |
| [6] | Diaz, J.I.: Nonlinear partial differential equations and free boundaries. Research Notes in Mathematics106. Boston: Pitman 1985 |
| [7] | Hess, P.: On the solvability of nonlinear elliptic boundary value problems. Indiana Univ. Math. J.25, 461-466 (1976) · Zbl 0329.35029 · doi:10.1512/iumj.1976.25.25036 |
| [8] | Kazdan, L., Warner, F.W.: Remarks on some quasilinear elliptic equations. Commun. Pure Appl. Math.28, 567-597 (1975) · Zbl 0325.35038 · doi:10.1002/cpa.3160280502 |
| [9] | de Mottoni, P., Schiaffino, A., Tesei, A.: Attractivity properties of nonnegative solutions for a class of nonlinear degenerate parabolic problems. Ann. Mat. Pura Appl.136, 35-48 (1984) · Zbl 0556.35083 · doi:10.1007/BF01773375 |
| [10] | Peletier, L.A., Tesei, A.: Global bifurcation and attractivity of stationary solutions of a degenerate diffusion equation. Adv. Appl. Math.7, 435-454 (1986) · Zbl 0624.35006 · doi:10.1016/0196-8858(86)90024-2 |
| [11] | Pozio, M.A., Tesei, A.: Support properties of solutions for a class of degenerate parabolic problems, Commun. Partial Differ. Equations12, 47-75 (1987) · Zbl 0629.35071 · doi:10.1080/03605308708820484 |
| [12] | Sattinger, D.H.: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J.21, 979-1000 (1972) · Zbl 0223.35038 · doi:10.1512/iumj.1972.21.21079 |
| [13] | Schatzman, M.: Stationary solutions and asymptotic behaviour of a quasilinear degenerate parabolic equation. Indiana Univ. Math. J.33, 1-30 (1984) · Zbl 0554.35064 · doi:10.1512/iumj.1984.33.33001 |
| [14] | Senn, S.: On a nonlinear elliptic eigenvalue problem with Neumann boundary conditions, with an application to population genetics. Commun. Partial Differ. Equations8, 1199-1228 (1983) · Zbl 0526.35067 · doi:10.1080/03605308308820300 |
| [15] | Spruck, J.: Uniqueness in a diffusion model of population biology. Commun. Partial Differ. Equations8, 1605-1620 (1983) · Zbl 0534.35055 · doi:10.1080/03605308308820317 |
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.
