A priori bounds for positive solutions of Kirchhoff type equations

Abstract

Let Ω be a bounded smooth domain in R N . Assume that 0 < α < 2 * −1 2 , a > 0, and b > 0. We consider the following Dirichlet problem of Kirchhoff type equationWhere 2 * = +∞ for N = 2, and 2 * = N +2 N −2 for N ≥ 3. Under suitable conditions of h(x, u, ∇u) (see (A), (H 1 ) and (H 2 ) in section 3), we get a priori estimates for positive solutions to problem (0.1). By making use of these estimates and the continuous method, we further get existence results for positive solution to problem (0.1) when 0 < p < 1, or 2α + 1 < p < 2 * . Effects of the term b||∇u|| 2α+1 2 on the solution set of problem (0.1) can be seen from an example given in section 2.

Positive solutions of inhomogeneous Kirchhoff type equations with indefinite data

“…Back to the Kirchhoff type equations (that is the case b > 0), it attracts more and more attentions in the recent years. See for example [6,32,12,11,33,34,26,39,35,44,47,10,15,20,45,17,37,28,43,30,46]. Most literatures available so far are concerning with ground state solutions for homogenous Kirchhoff equations.…”