The Boundary Value Problem For Singular Nonlinear Elliptic Equations And Its Eigenvalue Inequalities

We firstly introduce in Chapter 1 the history, background and present situation of the boundary problem for some nonlinear elliptic equations with singularity and the development of some variational inequalities.In chapter 2, the nontrivial solutions in space H₀²(B) of the Dirich-let problem for a biharmonic elliptic equation with the involving critical logarithm weightis studied. Where B is an open ball in R⁴, and p > 2.Firstly, the critical weight for the nonlinear biharmonic equation is defined, and then get the Hardy inequality with logarithm weightfor all nontrivial u(x) ∈ H₀²(B), if x ≠ x₀ and the dimension N = 4. By this inequality, we can prove that problem (1) satisfies the Mountain Pass geometry. Thus, the result is obtained: If λ > λ₀ = 1/4, then the problem (1) has no nontrivial solutions for all p ∈ (2, +∞) (including subcritical, critical, supercritical cases); If λ < λ₀, then the problem (1) has at least a positive solution.In chapter 3, the eigenvalue inequalities for biharmonic equations with two kinds of singularity are discussed. One kind of singularity belongs to L^∞(Ω), that iswhere λ > 0, a(x) ∈ L^∞(Ω) and a(x) > 0, Ω R^N(N ≤ 2) is a bounded domain with smooth boundary. As to problem (2), we get the inequality...