MULTIPLE POSITIVE SOLUTIONS OF A STURM-LIOUVILLE BOUNDARY VALUE PROBLEM WITH CONFLICTING NONLINEARITIES

作者：Feltrin Guglielmo^*

来源：Communications on Pure and Applied Analysis, 2017, 16(3): 1083-1102.

DOI：10.3934/cpaa.2017052

• Superlinear indefinite problems
• positive solutions
• Sturm-Liouville boundary conditions
• multiplicity results
• Leray-Schauder topological degree

We study the second order nonlinear differential equation
u" + Sigma(m)(i=1) alpha(i)a(i)(x)g(i)(u) - Sigma(m+1)(j=0) beta(j)b(j)(x)k(j)(u) = 0,
Where alpha(i beta j) > 0, a(j()x), b(j()x) are non-negative Lebesgue integrable functions de fi ned 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 beta(j) are sufficiently large, we prove the existence of at least 2(m-1) positive solutions for the SturmLiouville 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.

• 出版日期2017-5