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In this work, we prove existence and multiplicity of solutions to a Kirchhoff elliptic problem in the whole space ℝN. More specifically, we consider the following nonlocal elliptic problem: − m(∥∇u∥22)Δu + V (x)u = λa(x)|u|q−2u − 휃b(x)|u|p−2uin ℝN,u ∈ H1(ℝN), where N ≥ 3, the parameters λ,휃 > 0, 2 < 2(σ + 1) < q < p < 2∗ := 2N/(N − 2), σ ∈ (0, 2/(N − 2)) and a,b ∈ L∞(ℝN) with a(x),b(x) > 0 almost everywhere in ℝN. This type of problem contains the function m : ℝ+ → ℝ+ known as the Kirchhoff function given by m(t) = α1 + α2tσ with α1,α2 > 0 and t ∈ ℝ+. Under our assumptions the potential V : ℝN → ℝ and the nonlinearities can be sign changing functions. Hence, our main objective is to prove that the above problem has at least two distinct nontrivial solutions, one of them being a ground state, whenever λ ∈ (λ∗, +∞) for some suitable λ∗ > 0. The main idea is to use the minimization method in the Nehari manifold together with the nonlinear Rayleigh quotient. In our setting, the main difficulty is ensuring the existence of nontrivial solutions by using the Nehari method considering the Lagrange Multipliers Theorem. In other words, we study the case where the fibering map admits inflections points with λ,휃 > 0. Furthermore, for each λ ∈ (−∞,λ∗], we show a nonexistence result for our main problem. It is important to emphasize that λ∗ > 0 is sharp in order to find the existence and multiplicity of nontrivial solutions for our main problem. [ABSTRACT FROM AUTHOR]

The existence and multiplicity of solutions for a class of quasilinear elliptic problems are established for the type − Δ Φ u = f (u) in  Ω , u = 0 on  ∂ Ω , where Ω ⊂ ℝ N , N ≥ 2 , is a smooth bounded domain. The nonlinear term f : ℝ → ℝ is a continuous function which is superlinear at the origin and infinity. The function Φ : ℝ → ℝ is an N -function where the well-known Δ 2 -condition is not assumed. Then the Orlicz–Sobolev space W 0 1 , Φ (Ω) may be non-reflexive. As a main model, we have the function Φ (t) = (e t 2 − 1) / 2 , t ≥ 0. Here, we consider some situations where it is possible to work with global minimization, local minimization and mountain pass theorem. However, some estimates employed here are not standard for this type of problem taking into account the modular given by the N -function Φ. [ABSTRACT FROM AUTHOR]