Abstract
This chapter is devoted to the study of L p Lyapunov-type inequalities (\(1 \leq p \leq +\infty\)) for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in R N. In the case of Dirichlet conditions, it is possible to obtain analogous results in an easier way. We also treat the case of higher eigenvalues in the radial case, by using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions. It is proved that the relation between the quantities p and N∕2 plays a crucial role just to have nontrivial Lyapunov inequalities. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems.
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Cañada, A., Villegas, S. (2015). Partial Differential Equations. In: A Variational Approach to Lyapunov Type Inequalities. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-25289-6_4
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DOI: https://doi.org/10.1007/978-3-319-25289-6_4
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