We study the second order nonlinear differential equation u''+∑_i α_ia_i(x)g_i(u) − ∑j β_jb_j(x)k_j(u)=0, where α_i,β_j>0, a_i(x), b_j(x) are non-negative Lebesgue integrable functions defined in [0,L], and the nonlinearities g_i(s), k_j(s) are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation u''+a(x)u^p=0, with p>1. When the positive parameters β_j are sufficiently large, we prove the existence of at least 2^m-1 positive solutions for the Sturm-Liouville boundary value problems associated with the equation. The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets. Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.

Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities

Feltrin, Guglielmo
2017-01-01

Abstract

We study the second order nonlinear differential equation u''+∑_i α_ia_i(x)g_i(u) − ∑j β_jb_j(x)k_j(u)=0, where α_i,β_j>0, a_i(x), b_j(x) are non-negative Lebesgue integrable functions defined in [0,L], and the nonlinearities g_i(s), k_j(s) are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation u''+a(x)u^p=0, with p>1. When the positive parameters β_j are sufficiently large, we prove the existence of at least 2^m-1 positive solutions for the Sturm-Liouville boundary value problems associated with the equation. The proof is based on the Leray-Schauder topological degree for locally compact operators on open and possibly unbounded sets. Finally, we deal with radially symmetric positive solutions for the Dirichlet problems associated with elliptic PDEs.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11390/1149610
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