Existence Of Semilinear Sturm-liouville Boundary Value Problems Of Non-trivial Non-negative Solutions

In this paper, we consider the existence of nontrivially nonnegative solutions to theboundary value problemsWe will assume that the following conditions are satisfied:(H1)"Sturm-Liouville assumption":p∈C¹[a,b], q,h∈C[a,b],p(r) > 0 and h(r) > 0 for all r∈[a,b],α₀² +α₁² > 0,β₀² +β₁² > 0.(H2) f : [a,b]×[0,∞)â†'(-∞,∞) is continuous.(H3) f(r,0) = 0 for all r∈[a,b].(H4) There existδ0≥0 andδ1≥0 such thatf(r,z)≥-(δ₁z +δ₀) for all r∈[a,b] and z≥0.Letμ_* be the first eigenvalue of the problem(p(r)u')' + q(r)u +μh(r)u = 0 in (a,b), R₁u = R₂u = 0. (1.2)Our main results as follows.Theorem 1.1 Assume that (H1)–(H4) hold. Iforthen the problem (1.1) has at least one nontrivially nonnegative solution.