×

A note on a nonresonance condition for a quasilinear elliptic problem. (English) Zbl 0816.35031

The authors study sufficient conditions for nonresonance of the problem \[ -\text{div}(| \nabla u|^{p-2} \nabla u)= f(u)+ h(x) \quad (x\in \Omega), \qquad u=\varphi \quad (x\in \partial \Omega), \] i.e. existence of solutions for any given \(h\in L_ \infty (\Omega)\) and \(\varphi\in W_ p^{1-1/p} (\partial \Omega)\cap L_ \infty (\partial \Omega)\). Typically, these conditions are formulated in terms of the “interaction” of the asymptotic growth of the primitive of the nonlinearity \(f\), on the one hand, and the first eigenvalue of the above differential operator, on the other.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs

Keywords:

nonresonance
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anane, A., Simplicité et isolation de la première valeur propre du \(p\)-Laplacien avec poids, C. r. Acad. Sci. Paris, 305, 725-728 (1987) · Zbl 0633.35061
[2] Fernandez, M.; Omari, P.; Zanolin, F., The solvability of a semi-linear two point boundary value problem around the first eigenvalue, Diff. Integral Eqns, 2, 63-79 (1989) · Zbl 0715.34037
[3] Fonda, A.; Gossez, J. P.; Zanolin, F., On a nonresonance condition for a semilinear elliptic problem, Diff. Integral Eqns, 4, 945-951 (1991) · Zbl 0735.35054
[4] Deuel, J.; Hess, P., A criterion for the existence of solutions of nonlinear elliptic boundary value problems, Proc. R. Soc. Edinb., 74, 49-54 (1975) · Zbl 0331.35028
[5] Opial, Z., Sur les périodes des solutions le l’équation \(x″ + g(x) = 0\), Annls pol. Math., 10, 49-72 (1961) · Zbl 0096.29604
[7] Anane, A.; Gossez, J.-P., Strongly nonlinear elliptic problems near resonance: a variational approach, Communs partial diff. Eqns, 15, 1141-1159 (1990) · Zbl 0715.35029
[8] Otani, M., A remark on certain nonlinear elliptic equation, Proc. Fac. Sci. Tokai Univ., 19, 23-28 (1984) · Zbl 0559.35027
[9] Vo-Khac-Khoan, Distributions, analyse de Fourier, opérateurs aux dérivées partielles (1972), Vuibert · Zbl 0252.46093
[10] Tolksdorf, P., Regularity for a more general case of quasilinear elliptic equations, J. diff. Eqns, 51, 126-150 (1984) · Zbl 0488.35017
[11] Thelin, F. De, Local regularity properties for the solution of a nonlinear partial differential equation, Nonlinear Analysis, 6, 839-844 (1982) · Zbl 0493.35021
[12] Simon, J., Sur des équations aux dérivées partielles non linéaires, Thése (1977), Paris
[13] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
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.