Nonexistence Of Finite Energy Solution To A Quasilinear Elliptic Equation System With Singular And Gradient Terms

In this paper, we study the homogeneous Dirichlect boundary value problem to a system of quasi-linear elliptic p-Laplace equations with singular and gradient terms, and we obtain the nonexistence of finite energy solution. The background of the system a model derived from the study of the yield pseudo-plastic fluid. The system is coupled by the following two equations The mathematical model play an important role in many fields such as the study of rock and soil medium, oil exploration and extraction etc., so the study to the system is more valuable.The paper is divided into four sections. In the former part of the introduction,we first mention the background of plastic fluid model and the progress in studying the model. In addition, we make an emphasize explanation to the background and the value of the study of the system. In the latter part of the introduction, we introduce the problem we are studying: where p>1, q>1are constants, f(x), g(x) are nonnegative functions and f=0, g≠0, in addition, a(x, u), b(x, v) are weight functions which are measurable in Ω×(0,+∞) in the sense of the Lebesgue measure.The parameters (p, q) are medium indicators. In the case (p, q)>(2,2), the system reflects the property of dilatant fluid. When (p, q)<(2,2), the system reflects the property of pseudo plastic fluid. When (p, q)=(2,2), the system reflects the property of newtonian fluid.We consider the case when u-0and v=0, are the singular points of a(x, u), b(x, v), namely, lima(x, s)=+∞, limb(x, s)=+∞.In the second section, we introduce some basic knowledge to be used and make a brief introduction.The third section is devoted to the proof of our main result which is the following theorem.Theorem:Suppose that f∈L’(Ω)(r≥N/P),f≥0,f≠0,λ(f)/2≥1;g∈L’(Ω)(r≥N/q), g≥0,g≠0, λ(g)/2≥1. If there exist nonnegative functions m∈C((0,+∞),[0,+∞)), n∈C((0,+∞),[0,+∞)) satisfying such that(1)0≤m(s)≤a(x, s) a.e. x∈Ω. Vs>0, m(s)=0, s>1,(2)0≤n(s)≤b(x, s) a.e. x∈Ω. Vs>0, n(s)=0, s>1.Then the problem has no solution with finite energy. The A(f) in the above theorem is the first eigenvalues of the following problem and A(g) is the first eigenvalues of the following problemIn the study of the problem, we first define the following functions with the use of λ(f), and A(g).Then we construct the following test functions and prove the main result by a proof with contradiction and the use of the Poincare inequality.In the last section, we make a summary, and put forward some new problems.