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Perturbation from Dirichlet problem involving oscillating nonlinearities. (English) Zbl 1220.35030
The authors deal with the following perturbed Dirichlet problem \[ -\Delta u= f(x, u)+\lambda g(x,u)\quad\text{in }\Omega, \] \[ u= 0\quad\text{on }\partial\Omega, \] where \(\Omega\) is a bounded set in \(\mathbb{R}^d\) with smooth boundary, \(f,g: \Omega\times\mathbb{R}\to \mathbb{R}\) are given functions. The key role is played by the assumptions on \(f(x,\cdot)\) that \(f\) is allowed to change sign, uniformly with respect to \(x\), in any neighborhood of zero (respectively \(+\infty\)), which in turn implies an oscillating behaviour for its potential \(F(x,t)\), \(F(x,t):= \int^t_0 f(x,s)\,ds\). Under some suitable assumptions \(f\), \(g\) and \(\lambda\), the authors prove prove existence of at least \(k\) distinct weak solutions in \(W^{1,2}_0(\Omega)\), for every \(k\in\mathbb{N}\).

35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] Anello, G.; Cordaro, G., Infinitely many arbitrarily small positive solutions for the Dirichlet problem involving the p-Laplacian, Proc. roy. soc. Edinburgh sect. A, 132, 511-519, (2002) · Zbl 1064.35053
[2] Degiovanni, M.; Radulescu, V., Perturbation of nonsmooth symmetric nonlinear eigenvalue problems, C. R. acad. sci. Paris Sér. I, 329, 281-289, (1999)
[3] Li, S.J.; Liu, Z.L., Perturbations from symmetric elliptic boundary value problems, J. differential equations, 185, 271-280, (2002) · Zbl 1032.35066
[4] Liu, Z.L.; Su, J.B., Solutions of some nonlinear elliptic problems with perturbation terms of arbitrary growth, Discrete contin. dyn. syst., 10, 617-634, (2004) · Zbl 1205.35118
[5] Omari, P.; Zanolin, F., Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Comm. partial differential equations, 21, 721-733, (1996) · Zbl 0856.35046
[6] Saint Raymond, J., On the multiplicity of the solutions of the equations \(- \operatorname{\Delta} u = \lambda f(u)\), J. differential equations, 180, 65-88, (2002) · Zbl 1330.35279
[7] Ricceri, B., A general variational principle and some of its applications, J. comput. appl. math., 113, 401-410, (2000) · Zbl 0946.49001
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