Positive solutions for some generalized $ p $–Laplacian type problems

Abstract

In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem    −div(a(x, u, ∇u)) + At(x, u, ∇u) = f (x, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ R N is an open bounded domain, N ≥ 3, and A(x, t, ξ), f (x, t) are given functions, with At = ∂A ∂t , a = ∇ ξ A. To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami-Palais-Smale condition. Furthermore, if A(x, t, ξ) grows fast enough with respect to t, then the nonlinear term related to f (x, t) may have also a supercritical growth.

Existence of minimizers for some quasilinear elliptic problems

“…Furthermore, in Section 4, again by minimization arguments, we prove the existence of a positive solution of problem (GP ) under stronger hypotheses (see [1] and [9] for the super-p-linear case).…”

Multiple solutions for some symmetric supercritical problems

“…Different arguments can be found in [4] where, by using a sequence of truncated functionals, the authors prove that problem (1.4) with, e.g., p = 2, has at least one positive solution if (1.6) and the further condition 2(s + 1) < 2 * hold, which imply N < 6 (see [4,Theorem 2.1]). Differently from [4], here we use variational methods which exploit the interaction between two different norms and we do not require this additional restriction (see also [10] where, in the same setting of Theorem 1.1, the existence of at least one positive solution of problem (1.4) is proved). This paper is organized as follows.…”

“…10) implyJ (u) ≥ α 1 α 2 + α 3 µ |u| ps )|∇u| p dx − a |u| q dx − (η 3 + a 3 )|Ω|,(4.3) while from (3.24) and (4.1) it follows that Ω |u| ps )|∇u| p dx ≥ 1 (s + 1) p ℓ W,s (u) p . (4.4)…”