×

zbMATH — the first resource for mathematics

A Neumann problem with asymmetric nonlinearity and a related minimizing problem. (English) Zbl 0910.34035
Let be \(f\in C([0,1] \times\mathbb{R}, \mathbb{R})\), \(h\in L^2(0,1)\), \(\lim_{s\to-\infty} (f(x,s)/s) < {\pi^2 \over 4}\) uniformly in \(x\in [0,1]\), and \(\lim_{s\to- \infty} f(x,s)<-\int^1_0 h(x)dx <\lim_{s\to \infty} f(x,s)\) uniformly in \(x\in [0,1]\), then the problem \(-u''= f(x,u) +h(x)\), \(x\in(0,1)\), \(u'(0)= u'(1)=0\), has a weak solution. The proof is based on variational methods. A partial differential equation version of these problems is also considered.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ambrosetti, A.; Hess, P., Positive solutions of asymptotically linear elliptic eigenvalue problems, J. math. anal. appl., 73, 411-422, (1980) · Zbl 0433.35026
[2] Ambrosetti, A.; Prodi, G., On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. math. pura appl., 93, 231-247, (1973) · Zbl 0288.35020
[3] Ambrosetti, A.; Rabinowitz, P.H., Dual variational methods in critical point theory and applications, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[4] Amann, H.; Hess, P., A multiplicity result for a class of elliptic boundary value problems, Proc. roy. soc. edingburgh, 84, 145-151, (1979) · Zbl 0416.35029
[5] Dancer, E.N., Boundary-value problems for weakly nonlinear ordinary differential equations, Bull. austral. math. soc., 15, 321-328, (1976) · Zbl 0342.34007
[6] De Figueiredo, D.G., On superlinear elliptic problems with nonlinearities interacting only with higher eigenvalues, Rocky mountain J. math., 18, 287-303, (1988) · Zbl 0673.35027
[7] De Figueiredo, D.G.; Ruf, B., On a superlinear sturm – liouville equation and a related bouncing problem, J. reine angew. math., 421, 1-22, (1991) · Zbl 0732.34024
[8] De Figueiredo, D.G.; Solimini, S., A variational approach to superlinear elliptic problems, Comm. partial differential equations, 9, 699-717, (1984) · Zbl 0552.35030
[9] Kannan, R.; Ortega, R., Existence of solution ofxxgxptxxπ, Proc. amer. math. soc., 96, 67-70, (1986)
[10] Landesman, E.; Lazer, A., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203
[11] Lazer, A.; McKenna, P., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM review, 32, 537-578, (1990) · Zbl 0725.73057
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.