# zbMATH — the first resource for mathematics

Nonuniform nonresonance at the first eigenvalue of the $$p$$-Laplacian. (English) Zbl 0876.35039
The author considers the Dirichlet boundary value problem on the interval $$[0,\omega]$$ for the equation $(\phi_p(u'(t)))'= f(t,u(t),u'(t)),\tag{*}$ where $$u(t)=(u_1(t),...,u_m(t))$$, $$\phi_p(u_1,...,u_m)=(|u_1|^{p-2}u_1,...,|u_m|^{p-2}u_m)$$, $$1<p<\infty$$, $$f$$ is an $$L^1$$-Caratheodory mapping. He also studies the associated eigenvalue problem, $$-(\phi_p(u'))'=\lambda\phi_p(u)$$, $$u(0)=u(\omega)=0$$. Under some additional conditions on $$f$$, involving the first eigenvalue $$\lambda_1$$, it is proved that the equation (*) has at least one solution $$u\in C^1$$ such that $$\phi_p(u')$$ is absolutely continuous.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 34B15 Nonlinear boundary value problems for ordinary differential equations
##### Keywords:
existence; first eigenvalue
Full Text:
##### References:
 [1] Mawhin, J., Boundary value problems at resonance for vector second order nonlinear ordinary differential equations, (), 241-249 [2] Del Pino, M.; Elgueta, M.; Manásevich, R., A homotopic deformation along p of a Leray-Schauder degree result and existence for (¦u′p−2 u′)′ + ƒ(t, u) = 0, u(0) = u(T) = 0, p > 1, J. diff. eqns, 80, 1-13, (1989) · Zbl 0708.34019 [3] Fučik, S.; Nečas, J.; Souček, J.; Souček, V., Spectral analysis of nonlinear operators, () · Zbl 0268.47056 [4] Boccardo, L.; Dràbek, P.; Giachetti, D.; Kučera, M., Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear analysis, 10, 1083-1103, (1986) · Zbl 0623.34031 [5] Njoku, F.I.; Zanolin, F., Positive solutions for two-point BVPs: existence and multiplicity results, Nonlinear analysis, 13, 1329-1338, (1989) · Zbl 0704.34020 [6] Manásevich, R.; Zanolin, F., Time-mappings and multiplicity of solutions for the one-dimensional p-Laplacian, Nonlinear analysis, 21, 269-291, (1993) · Zbl 0792.34021 [7] Mawhin, J.; Ward, J.R., Nonresonance and existence for nonlinear elliptic boundary value problems, Nonlinear analysis, 5, 677-684, (1981) · Zbl 0475.35047 [8] Mawhin, J.; Ward, J.R.; Willem, M., Variational methods and semilinear elliptic equations, Arch. ration. mech. analysis, 95, 269-277, (1986) · Zbl 0656.35044 [9] Gupta, C.P.; Kwong, Y.G., Nonresonance conditions for the strong solvability of a general elliptic partial differential operator, Nonlinear analysis, 17, 613-625, (1991) · Zbl 0763.35039 [10] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025 [11] Granas, A.; Guenther, B.R.; Lee, J.W., Some general existence principles in the Carathéodory theory of nonlinear differential systems, J. math. pures appl., 70, 153-196, (1991) · Zbl 0687.34009
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.