In this paper we consider the nonlinear Choquard equationu) is of critical growth due to the Hardy-Littlewood-Sobolev inequality and G(x, u) =ˆu 0 g(x, s)ds. Firstly, by assuming that the potential V (x) might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami [7], we also study the existence of high energy solution by using a global compactness lemma for the nonlocal Choquard equation. 2010 Mathematics Subject Classification. 35J20, 35J60, 35A15.

In this paper, we consider the existence of least action nodal solutions for the quasilinear defocusing Schrödinger equation in [Formula: see text]: [Formula: see text] where [Formula: see text] is a positive continuous potential, [Formula: see text] is of subcritical growth, [Formula: see text] and [Formula: see text] are two non-negative parameters. By considering a minimizing problem restricted on a partial Nehari manifold, we prove the existence of least action nodal solution via deformation flow arguments and [Formula: see text]-estimates.

This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem { div ϕfalse(false|∇ufalse|false)∇u=afalse(xfalse)ffalse(ufalse)inΩ,u≥0inΩ,u=∞on∂Ω,where either Ω⊂boldRN with N≥1 is a smooth bounded domain or Ω=boldRN. The function ϕ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, generalized Newtonian fluids, and in quantum physics. The proofs are based on comparison principle, variational methods and topological arguments on the Orlicz–Sobolev spaces.

We deal with existence of large solutions of ∆

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