Positive solutions of quasilinear elliptic equations in exterior domains

Abstract

Our main purpose is to prove the existence of a positive solution of the quasilinear second order elliptic differential equation (1) below in an exterior domain in Euclidean space Rn, n 3 2. More explicitly, if (1) has a positive subsolution w and a positive supersolution u such that W(X) < V(X) throughout an exterior domain G, (notation in Section 2), Theorem 3.3 establishes the existence of a solution u of (1) in G, such that w(x) < U(X) < a(z) for all x E G, . An analogous theorem for bounded domains due to Nagumo [7] is employed in Lemma 3.1 to construct a sequence of positive solutions of (1) in annular domains G(a, a + j), j = 1, 2 ,.... We then prove in Lemma 3.2 and Theorem 3.3 that this sequence converges to a positive solution of (1) in G, by means of interior Schauder estimates, D-space estimates, and a priori interior estimates on the gradient of a solution of (1) in a bounded domain. Theorem 3.3 is applied in Section 4 to give sufficient conditions for a quasilinear SchrBdinger equation (8) to have a solution in an exterior domain in Rn satisfying 0 < U(X) < 1 x 10, 4 = 2 n + E, for arbitrary E in (0, n 2), n 3 3; and a similar result is given for IZ = 2. Section 5 indicates the procedure for reduction of nonoscillation problems for (8) to corresponding problems for quasilinear ordinary differential equations or inequalities. In particular, criteria are given for the existence of a bounded positive solution of the generalized Emden-Fowler equation in some exterior domain. A necessary and sufficient