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Three solutions for a perturbed Dirichlet problem. (English) Zbl 1141.35030
Summary: We prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem: $\begin{gathered} -\Delta u= f(x,u)+\lambda g(x,u)\qquad\text{in }\Omega,\\ u= 0\qquad\text{on }\partial\Omega.\end{gathered}$ where $$\Omega\subset\mathbb{R}^N$$ is an open bounded set with smooth boundary $$\partial\Omega$$ and $$k\in\mathbb{R}$$. Under very mild conditions on $$g$$ and some assumptions on the behaviour of the potential of $$f$$ at $$0$$ and $$+\infty$$, our result assures the existence of at least three distinct solutions to the above problem for $$\lambda$$ small enough. Moreover such solutions belong to a ball of the space $$W^{1,2}_0(\Omega)$$ centered in the origin and with radius not dependent on $$\lambda$$.
##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35J20 Variational methods for second-order elliptic equations
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##### References:
 [1] G. Anello, Existence and multiplicity of solutions to a perturbed Neumann problem, Math. Nachr. (in press) · Zbl 1142.35014 [2] Anello, G.; Cordaro, G., Perturbation from Dirichlet problem involving oscillating nonlinearities, J. differential equations, 234, 80-90, (2007) · Zbl 1220.35030 [3] Cordaro, G., Multiple solutions to a perturbed Neumann problem, Studia math., 178, 2, 167-175, (2007) · Zbl 1387.35154 [4] Fan, X.; Deng, S.-G., Remarks on ricceri’s variational principle and applications to $$p(x)$$-Laplacian equations, Nonlinear anal., 67, 11, 3064-3075, (2007) · Zbl 1134.35035 [5] Faraci, F., Multiple solutions for two nonlinear problems involving the $$p$$-Laplacian, Nonlinear anal., 63, e1017-e1029, (2005) · Zbl 1224.35152 [6] Li, S.J.; Liu, Z.L., Perturbations from symmetric elliptic boundary value problems, J. differential equations, 185, 271-280, (2002) · Zbl 1032.35066 [7] Ricceri, B., Sublevel sets and global minima of coercive functional and local minima of their perturbations, J. nonlinear convex anal., 52, 157-168, (2004) · Zbl 1083.49004 [8] Ricceri, B., A multiplicity theorem for the Neumann problem, Proc. amer. math. soc., 134, 1117-1124, (2006) · Zbl 1134.35345 [9] Willem, M., Minimax theorems, (1996), Birkhauser Boston · Zbl 0856.49001
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