Existence Of Solutions Of A Class Of Quasilinear Elliptic Equations With Critical Sobolev Exponent

In this paper, we consider the existence of solutions for the quasilinear elliptic equation with Dirichlet boundary value conditionsas followswhereΩ(?) R^N(N≥3) is a bounded domain with smooth boundary, 2 < p < N, p^* := N_p/N-p is critical Sobolev exponent, and -â–³_pu = -div(|â–½u|^p-2â–½u) is the p-Laplacian of u.When A(u)≡0, the equation changes to -â–³_pu=|u|^p*-2u+λ|u|^p-2u+g(x,u)+h(x).and it has a variational structure. Brezis and Nirenberg considerded the following problem in [1]where a(x)∈L^∞(Ω), g(x,u) satisfies some smooth and structure conditions, and the minimum eigenvalueαof operator -â–³-a(x) is positive, namely∫_Ω(|▽φ|²-a(x)φ²)dx≥α∫_Ωφ²dx,(?)φ∈W₀^1,2(Ω),α>0.By using the mountain pass lemma, they proved that the problem(2) has solutions, if there exists some v≥0,v(?) 0,v∈W₀^1,2 (Ω), such that(?)I(tv)<1/NS^2/N (3)whereTherefore, we only need to verify whether (3) holds, to deal with the solvability of problem (2). Specialy, if a(x)≡λ, g(x,u) = 0, then problem (2) changes toThey proved that ifλ∈(0,λ₁), then (3) follows, thereby (5) has solutions, whereλ₁ is the first eigenvalue of -â–³in W₀^1,2 (Ω), namelyFurthermore, the authors gave some sufficient conditions under which(3) follows. When 1 < p²â‰¤N, by using concentration compactness lemma, they proved the problemhas nontrivial solution when A∈(0,λ₁), whereλ₁ is the first eigen-value of -â–³_p in W₀¹,2 (Ω), namelyBy using concentration compactness lemma, Zhu extended the problem(2) to the case 2≤p < N in [3], and the existence of nontrivial solutions for the problem was obtained. Be different from linear equations, the uniqueness of solutions for nonlinear equations does not always hold. So, it is natural to consider the multiplicity of solutions for nonlinear equations. In [4] the author considerded the following problemBy applying concentration compactness lemma, mountain pass lemma,via studing the critical sequence, Azorero and Alonso proved the problem (7) has solutions. They also proved that there are infinite many solutions for probelm (7) in the case 1 < q < p. In [5] the author considered the subcritical problemwhere l*. They proved that there exists a sequence of solutions with the corresponding energy goes to infinity. In [6] the author proved that there exists someλ₀ > 0, such that for allλ∈(0,λ₀), the following problem has at least two positive solutionsIf we replace the operators -â–³and -â–³_p with -â–³-μu/|x|² and-â–³_p-μ|u|^p-2u/|x|^p in (5) and (6), the equations have singularity. In [7] Jannelli considered the problemHe proved that the problem (9) has solutions if the following minimumvalue can be achievedIn [8] Ruiza and Willem proved that ifμ<μ- 1 = (N-2)²/4-1,λ∈(0,λ₁(μ)), then the value of (10) can be achieved, consequently the problem (9) has positive solutions whereIn [9] the author extended the conclusion to the caseλ> 0, however theμneeded to be strengthened toμ∈[0, (N-2/2)² - (N+2/N)²). In [10] the author extended the conclusion of [9] to the case when the the term |u|^p*-2u has variable coefficient. The author considered the problem They proved that if Q(x) is continue and satisfies some structure conditions, then the conclusion of [9] also holds. Under some weaker conditions the author proved the problem (11) has nontrivial solutionin [11].When A(u) (?) 0, problem (1) no longer has variational structure,so the vriational method does not work and there are far more scarce investigation methods and conclusions than the case A(u) (?) 0 be. In [12], by using topology degree theory and global implicit function theorem, the author proved that the problem (1) has nontrivial solution when p = 2, A (u) = cu ( c is some constant vector), g(x,u) (?) 0, h(x) (?) 0, and they also had obtained some bifurcation conclusion.In this paper, we study the existence of solutions of the problem (1), by using the theory of generalized Leray-Schauder topology degree together with the a prior estimates on solutions.