Existence And Multiplicity Of Solutions For φ-Laplacian Equations

In recent year, wide attention has been paid to the study differential equations with nonlinear operators, and the existence of positive solutions for the various boundary value problems has been obtained. In some papers, the authors studied these differential equations governed by p-Laplacian, that isφp(s)=｜s｜p-2 with p> 1. (see [16]-[34]). In other papers, the researchers studied the more general operatorφ-Laplacian. (see [12]-[15]). In the above references, the researchers studied these differential equations governed by p-Laplacian orφ-Laplacian operators by employing the topological degree theory, Leggett-Williams'fixed-point theorem and so on, and obtained the existence and multiplicity of solutions. It is necessary to list these papers ([5], [6], [8], [9], [13]) which are helpful to this paper.In paper [5], the author used the fixed point index method, and obtained the existence of at least one or at least two positive solutions for the third-order Sturm-Liouville boundary value problem with p-Laplacian. In Chapter one, we apply the method to the following third-order boundary value problem withφ-Laplacian: whereλ≥0,f,φsatisfies the following conditions:(A1)f:[0,1] x Râ†'R+is continuous, where R+= [0,∞);(A2)φ: (-Ï„,)â†'R is an increasing homeomorphism,φ(0)= 0.Using the method in paper [5], we obtain the following theorems:Theorem 1.3.1. Assume (A1), (A2) hold. In addition, there exist constants b, c> 0 such that b< min{m/ι,σ}c, andφ(mc)> 4φ(ιb),mc<Î¥,0<σ< 1. If the following assumptions hold:(H1) f(t,x)≥4φ(ιb) for 1/4≤t≤3/4,b≤x≤b/<σ;(H2) f(t, x)≤φ(mc) for 0≤t≤1,0≤x≤c, where Then the problem (1.1.1) has at least one solution u* with‖u*‖≤c and (?) u*(t)> b.Theorem 1.3.2. Assume (A1), (A2) hold. If the following conditions are satisfied: (H4) there exists a constant p> 0 such that f(t, x)<(mp),0≤t≤1,0≤x≤p, where m,1 is given as in (H2), (Hi). Then the problem (1.1.1) has at least two positive solutions u1 and u2, such that 0<‖u1‖< P<‖u2‖.In [13], the author used Leray-schauder degree theory to discuss a kind ofφ-Laplacian equation and obtained various existence results. In [6], the author study the existence of multiple positive solutions for one-dimensional p-Laplacian equation, by using Avery and Peterson theorem. Motivated by these papers and their results, in chapter two, we use the method to discuss the followingφ-Laplician equation boundary value problem: where f,φsatisfies the following conditions:(B1)f:[0, l]×R×Râ†'R+is continuous, where R+= [0,∞);(B2)φ:(-d, d)â†'R is an increasing homeomorphism,φ(0)= 0.(B3) g:Râ†'R+is continuous, and k≥1, g(Ï…)≤k｜υ｜,υ∈R.We obtain the following theorem:Theorem 2.3.1. Assume (B1), (B2) hold. Let 0< a< b≤δd/(k+1), and suppose that f satisfies the following conditions:(H5) f(t,u,Ï…)≤φ(d/2) for (t,u,Ï…)∈[0,1]×[0, (k+l)d]×[-d,0];(H6) f(t, u,Ï…)>φ(b/δ)/(1-δ) for (t, u,Ï…)∈[0,1-δ)×[b, b/δ]×x [-d,0];(H7) f(t, u,Ï…)<φ(a/(l+k)) for (t, u,Ï…)∈[0,1]×[0, a]×[-d,0]. then the boundary value problem (2.1.1) has at least three nonegative solutions u1,u2,u3, such that...