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### The Fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian

 Citace: DEL PINO, M., DRÁBEK, P., MANÁSEVICH, R. The Fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian. Journal of Differential Equations, 1999, roč. 1, č. 2, s. 386-419. ČLÁNEK eng The Fredholm alternative at the first eigenvalue for the one dimensional p-Laplacian 1999 Manuel Del Pino , Pavel Drábek , Raul Manásevich In this work we study the range of the operator u\mapsto (|u'|^{p-2}u')'+\lambda_1|u|^{p-2}u, u(0)=u(T)=0,p>1. We prove that all functions h\inC^1[0,T] satisfying \int^T_0 h(t)\sin_p(\pi_pt/T)dt=0 lie in the range, but that if p\neq2 and h\equiv0 the soltion set is bounded. Here sin(\pi_pt/T)is a first eigenfunction associated to \lambda_1. We also show that in this case the associated energy functional u\mapsto(1/p) \int^T_0|u'|^p-(\lambda_1/p) \int^T_0|u|^p+\int^T_0hu is unbounded from below if 1<2 and bounded from below (with a global minimizer) if p>2 on W^{1,p}_0 (0,T)(\lambda_1 corresponds precisely to the best constant in the L^p-Poincaré inequality). Moreover, we show that unlike the linear case p=2, for p\neq2 the range contains a nonempty opn set in L^{\infty}(0,T).

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