Multiple nontrivial solutions for nonlinear eigenvalue problems

Motreanu, D.; Motreanu, V.; Papageorgiou, N.

doi:10.1090/S0002-9939-07-08927-7

Multiple nontrivial solutions for nonlinear eigenvalue problems
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by D. Motreanu, V. V. Motreanu and N. S. Papageorgiou PDF

Proc. Amer. Math. Soc. 135 (2007), 3649-3658 Request permission

Abstract:

In this paper we study a nonlinear eigenvalue problem driven by the $p$-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

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