Integral Method And The One-dimensional P-laplacian Boundary Value Problem, Positive Solution

In this article the existence and nonexistence are considered for the p-Laplacianboundary value problemswhere p > 1,α≥0, andλ> 0 is a positive parameter. Assume that the followingconditions are satisfied:(A0) f : (0,+∞)â†'(-∞,+∞) is continuous, and there exists aμ^* > 0 such thatf(μ^*) = 0, f(u) > 0 for u∈(0,μ?) and f(u) < 0 for u∈(μ^*,+∞).(A1) For the case of T_μ* < +∞, there exists a 0 <ε<μ^* such that f is decreasingon (μ^* -ε,μ^*), where-1/pLetandwhere 0≤w≤r, r∈Λ.When T_μ* < +∞, denotewhere w_α^*∈[0,μ^*) be given by the equationThe main results of this paper as follows.Theorem 1.1 Assume that T_μ* = +∞.(1) The problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t)≥μ^*. (2) If limzâ†'0[f(z)/zp-1] = 0, then there exists aλ₁ > 0 such that(2-i) the problem (1.1) has two positive solutions u1,u2 which satisfiesforλ>λ₁.(2-ii) the problem (1.1) has a positive solution which satisfies maxt∈[0,1] u(t) <μ^*forλ=λ₁.(2-iii) the problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) <μ? forλ<λ₁.(3) If limzâ†'0[f(z)/zp-1] = c where 0 < c < +∞, then there exists aλ₁ > 0 suchthat(3-i) the problem (1.1) has a positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ>λ₁.(3-ii) the problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ<λ₁.(4) If limzâ†'0[f(z)/zp?1] = +∞, then the problem (1.1) has a positive solution whichsatisfies maxt∈[0,1] u(t) <μ? forλ> 0.Theorem 1.2 Assume that Tμ? < +∞.(1) The problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) >μ?.(2) The problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) =μ?forλ<λα?.(3) The problem (1.1) has a unique positive solution which satisfies maxt∈[0,1] u(t) =μ? forλ≥λα?.Theorem 1.3 Assume that Tμ? < +∞.(1) If limzâ†'0[f(z)/zp?1] = 0, then there exists a 0 <λ₁≤λα? such that(1-i) the problem (1.1) has a positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ>λ₁.(1-ii) the problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ<λ₁. (2) If limzâ†'0[f(z)/zp?1] = c where 0 < c < +∞, then there exists a 0 <λ₁≤λα?≤λ₂ < +∞such that(2-i) the problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ<λ₁ orλ>λ₂.(2-ii) the problem (1.1) has a positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ₁ <λ<λ₂.(3) If limzâ†'0[f(z)/zp?1] = +∞, then there exists aλ₁≥λα? > 0 such that(3-i) the problem (1.1) has no positive solution which satisfies maxt∈[0,1] u(t) <μ?forλ>λ₁.(3-ii) the problem (1.1) has a positive solution which satisfies maxt∈[0,1] u(t) <μ?for 0 <λ<λ₁.