A UNIVERSAL LOWER BOUND OF EXPONENTIAL INTEGRALS OF SOLUTIONS OF [xxx]
We prove that for any solutions of the nonlinear elliptic equation Au + f ( x , « ) = 0, f(x, ti) < a e " " (α, β > 0) on R 2 , the integral / + cannot J*? be a small number less than a positive constant C* independent of Μ , provided that / e is finite. A simple example (in §2) shows that the finiteness condition of e cannot be removed. AMS subject classification (1990). Primary 35J60; Secondary 35P30 Par t ly supported by the Grants-in-Aid for Scientific Research, the Ministry of Education, Science and Culture, Japan. 390 NOBUMITSU NAKAUCHI 1. I n t r o d u c t i o n In this note we are concerned with solutions of a nonlinear elliptic equation (1) Au + f(x, u) = 0 on R 2 , where (2) / ( x , u) < ote0" (α, β > 0). A typical example is the equation of prescribed Gaussian curvature Au Κχ{χ) + K2(x)e2u = 0 with nonnegative Κ ι and bounded on R2 . (See Kazdan-Warner [2].) This equation describes the pointwise conformai deformation gì = e 2 ugi of a given metric g\ on R 2 , where K\ , Κ2 are the Gaussian curvatures of gi , , respectively. The function u is called to be a weak solution of the equation (1) if the following two conditions hold: (а) u € L[0c(R2) > where L J ^ ( R 2 ) denotes the Sobolev space of functions whose derivatives belong to Lf 0 C (R 2 ) . (Then the Sobolev inequality implies u G Lfo c(R 2) for any ρ > 0.) (б) The function u satisfies (3) / Vw · Vy> + / f{x, υ)φ = 0 J R2 J R2 for any ψ € C ~ ( R 2 ) , where C£J°(R2) denotes the space of all C°°-functions with compact support in R 2 . In the study of the nonlinear eigenvalue problem f Δ « (4) { „ = + Ae" = 0 in Ω, 0 on dSi , the quantity Σ(ω, λ) = λ / e" plays an important role, where Ω is a domain in J Ω R 2 . (See Suzuki and Nagasaki [4] [8].) In the case that Ω is a disk, Σ defines a bijective correspondence between the totality of solutions (λ, u) of (4) and the interval [0, 4TT) ; i.e., Σ is a nonlinear faithful character. Considering these circumstances, we are interested in the family of the characters Σ(«, 7, δ) = y f e6" J Β2 A UNIVERSAL LOWER BOUND 391 for weak solutions « of (1). To begin with, we want to study what subset of [0, oo) the values of such integrals as I eP lie on, when u runs weak solutions of (1). In J R 2 this note we prove that / e " + e ) u cannot be any value of a neighborhood of 0, JV when u runs all weak solutions of (1) satisfying / e" J R 2 < oo : Theorem. For any ε > 0 , there exists a positive constant C* , depending only on ε, β, such that for any weak solution u of (1) satisfying (*) / e" J B = < OO,