A priori estimates for elliptic equations with reaction terms involving the function and its gradient

Abstract

We study local and global properties of positive solutions of −∆u = u p +M |∇u| q in a domain Ω of R N , in the range min{p, q} > 1 and M ∈ R. We prove a priori estimates and existence or non-existence of ground states for the same equation.

Radial solutions of scaling invariant nonlinear elliptic equations with mixed reaction terms

“…In the case N = 1 the result is already proved in [14]. The nonexistence of a ground state, not necessarily radial for M > −µ * is proved in [1] and independently in [7] with a different method. In the radial case it was obtained much before in the case N = 1 in [14] and then by Fila and Quittner [16] who raised the question whether the condition −μ min < M < 0 is optimal for the non-existence of radial ground state.…”

“…A general survey with several open problems can be found in [20]. In the non radial case an important contribution dealing with a priori estimates of local positive solutions of (1.2) and existence or non-existence of entire positive solution in R N is due to the authors [7]. In this paper we complete the results of [7] in presenting a quite exhaustive study of the radial solutions of (1.1) for any real number M .…”

“…Next we study the case where M is large enough. We have already proved in [7] that for any p > 1, there exists M † = M † (N, p) > 0 (see introduction) such that if M > M † there exists no ground state, radial or non-radial. In the radial case we have a more precise result.…”

A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

“…by Bidaut-Véron, García-Huidobro and Véron, see [6,7]. By using a delicate combination of refined Bernstein techniques and Keller-Osserman estimate, they obtained a series of a priori estimates for any positive solution of (1.9) in arbitrary domain Ω of R N in the case p > 1, q ≥ 2p p+1 and M > 0 ([6, Theorems A, C, D]).…”

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In this article we study local and global properties of positive solutions of −∆mu = |u| p−1 u + M |∇u| q in a domain Ω of R N , with m > 1, p, q > 0 and M ∈ R. Following some ideas used in [6,7], and by using a direct Bernstein method combined with Keller-Osserman's estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin's classical results.

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