Existence of ground states and free boundary problems for quasilinear elliptic operators

Abstract

We prove the existence of non-negative non-trivial solutions of the quasilinear equation Δmu + f(u) = 0 in ℝn and of its associated free boundary problem, where Δm denotes the m-Laplace operator. The nonlinearity f(u), defined for u > 0, is required to be Lipschitz continuous on (0,∞), and in L1 on (0, 1) with u 0 f(s) ds < 0 for small u > 0; the usual condition f(0) = 0 is thus completely removed. When n > m, existence is established essentially for all subcritical behavior of f as u→∞, and, with some further restrictions, even for critical and supercritical behavior. When n = m we treat various exponential growth conditions for f as u → ∞, while when n < m no growth conditions of any kind are required for f. The proof of the main results moreover yield as a byproduct an a priori estimate for the supremum of a ground state in terms of n, m and elementary parameters of the nonlinearity. Our results are thus new and unexpected even for the semilinear equation Δu + f(u) = 0. The proofs use only straightforward and simple techniques from the theory of ordinary differential equations; unlike well known earlier demonstrations of the existence of ground states for the semilinear case, we rely neither on critical point theory [6] nor on the Emden-Fowler inversion technique [2,3].