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

2016/07/28 by Feltrin, Guglielmo
#34B15 #34B18 #47H11 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1607.08365

Abstract

We study the second order nonlinear differential equation u"+ ∑_i=1^m α_i a_i(x)g_i(u) - ∑_j=0^m+1 β_j b_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 \mathopen[0,L\mathclose], 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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