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

DOI10.3934/CPAA.2017052zbMATH Open1365.34046arXiv1607.08365OpenAlexW2963981597MaRDI QIDQ519553

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Publication date: 5 April 2017

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Abstract: We study the second order nonlinear differential equation �egin{equation*} u"+ sum_{i=1}^{m} alpha_{i} a_{i}(x)g_{i}(u) - sum_{j=0}^{m+1} �eta_{j} b_{j}(x)k_{j}(u) = 0, end{equation*} where $$, $a_i(x),b_j(x)$ are non-negative Lebesgue integrable functions defined in $mathopen[0,Lmathclose]$, and the nonlinearities $g_i(s),k_j(s)$ are continuous, positive and satisfy suitable growth conditions, as to cover the classical superlinear equation $$, with $p>1$. When the positive parameters $$ 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.

Full work available at URL: https://arxiv.org/abs/1607.08365

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