×

Existence and uniqueness for a class of quasilinear elliptic boundary value problems. (English) Zbl 1042.34045

The authors prove the existence and uniqueness of positive solutions for the quasilinear boundary value problem \[ (r^{N-1}\phi(u'))'=-\lambda r^{N-1}f(u),\quad u'(0)=u(1)=0, (1) \] where \(\phi(x)=| x| ^{p-2}x\) and \(\lambda\) is a real parameter. Under the assumptions \( f:[0,\infty)\to \mathbb{R}\) is continuous, of class \(C^1\) on \((0,\infty)\), nondecreasing, with \[ \lim_{x\to \infty} f(x)>0,\quad \lim_{x\to \infty}{f(x)\over x^{p-1}}=0, \] the existence of a positive solution is proved.
If \(f(0)>0\), then a positive solution exists for all \(\lambda >0\) and if \(f(0)\leq 0\), then a positive solution exists for \(\lambda\) large. If moreover \[ \liminf_{x\to 0^+}{f(x)\over x^{p-1}}\not= 0, \quad \limsup_{x\to \infty}{xf'(x)\over f(x)}<p-1, \quad \limsup_{x\to 0^+}xf'(x)<\infty, \] then the uniqueness result for (1) is valid.
Note, that \(f(x)\over x^{p-1}\) may not be decreasing on \((0,\infty)\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 620-709 (1976) · Zbl 0345.47044
[2] Angenent, S. B., Uniqueness of the solution of a semilinear boundary value problem, Math. Ann., 272, 129-138 (1985) · Zbl 0576.35044
[3] F. Brock, Radial Symmetry for Nonnegative Solutions of Semilinear Elliptic Equations Involving the P-Laplacian, Progress in Partial Differential Equations, Vol. 1, Pitman Research Notes in Mathematics Series, Vol. 383, Longman Scientific & Technical, Harlow, 1997, pp. 46-57.; F. Brock, Radial Symmetry for Nonnegative Solutions of Semilinear Elliptic Equations Involving the P-Laplacian, Progress in Partial Differential Equations, Vol. 1, Pitman Research Notes in Mathematics Series, Vol. 383, Longman Scientific & Technical, Harlow, 1997, pp. 46-57. · Zbl 0920.35051
[4] Castro, A.; Hassanpour, M.; Shivaji, R., Uniqueness of nonnegative solutions for a semipositone problem with concave nonlinearity, Comm. Partial Differential Equations, 20, 1927-1936 (1995) · Zbl 0836.35049
[5] Dancer, E. N., On the number of positive solutions of weakly nonlinear elliptic equations when a parameter is large, Proc. London Math. Soc., 53, 429-452 (1986) · Zbl 0572.35040
[6] Drabek, P., Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Mathematics Series, Vol. 232 (1992), Longman Scientific & Technical: Longman Scientific & Technical Harlow · Zbl 0753.34002
[7] Erbe, L.; Tang, M., Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations, 133, 179-202 (1997) · Zbl 0871.34023
[8] Erbe, L.; Tang, M., Uniqueness theorems for positive radial solutions of quasilinear elliptic equations in a ball, J. Differential Equations, 138, 351-379 (1997) · Zbl 0884.34025
[9] Garcia-Huidobro, M.; Manasevich, R.; Schmitt, K., Positive radial solutions of quasilinear elliptic partial differential equations in a ball, Nonlinear Anal., 35, 175-190 (1999) · Zbl 0924.35047
[10] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020
[11] Guo, Z. M., Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations, Appl. Anal., 47, 173-190 (1992)
[12] Guo, Z. M.; Webb, J. R.L., Uniqueness of positive solutions for quasilinear elliptic equations when a parameter is large, Proc. Roy. Soc. Edinburgh, 124A, 189-198 (1994) · Zbl 0799.35081
[13] Hai, D. D., Uniqueness of positive solutions for a class of semilinear elliptic systems, Nonlinear Anal., 52, 595-603 (2003) · Zbl 1022.35013
[14] D.D. Hai, K. Schmitt, On radial solutions of quasilinear boundary value problems, Topics in Nonlinear Analysis, Progress in Nonlinear Differential Equations and their Applications, Vol. 35, Birkhauser, Basel, 1999, pp. 349-361.; D.D. Hai, K. Schmitt, On radial solutions of quasilinear boundary value problems, Topics in Nonlinear Analysis, Progress in Nonlinear Differential Equations and their Applications, Vol. 35, Birkhauser, Basel, 1999, pp. 349-361. · Zbl 0918.34027
[15] Lin, S. S., On the number of positive solutions for nonlinear elliptic equations when a parameter is large, Nonlinear Anal., 16, 283-297 (1991) · Zbl 0731.35039
[16] Sakaguchi, S., Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14, 403-421 (1987) · Zbl 0665.35025
[17] K. Schmitt, On boundary value problems for quasilinear elliptic equations, Differential Equations with Applications to Biology (Halifax, NS, 1997), Fields Institute Communication, Vol. 21, American Mathematical Society, Providence, RI, 1999, pp. 419-427.; K. Schmitt, On boundary value problems for quasilinear elliptic equations, Differential Equations with Applications to Biology (Halifax, NS, 1997), Fields Institute Communication, Vol. 21, American Mathematical Society, Providence, RI, 1999, pp. 419-427. · Zbl 0922.35059
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.