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In this paper, we study the nonexistence of solutions to the quasilinear Schrödinger equations: (1) − Δ p u + V (x) | u | p − 2 u − Δ p (| u | 2 α) | u | 2 α − 2 u = g (x , u) , x ∈ R N , where p-Laplacian operator Δ p u = d i v (| ∇ u | p − 2 ∇ u) , 1 < p < N and α > 1 / 2 is a parameter. Under suitable conditions on g (x , u) , the nonexistence of solutions to Equation (1) is investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions to a class of singular quasilinear elliptic problems − d i v (| x | − a p | ∇ u | p − 2 ∇ u) + V (x) | u | p − 2 u = λ h (x) | u | q − 2 u + H (x) | u | m − 2 u , x ∈ R N u (x) > 0 , x ∈ R N , u ∈ X ≡ D a 1 , p (R N , | x | − a p) ∩ L p (R N , V) , where 1 < p < N , − ∞ < a < N − p p , λ > 0 , 1 < p < q < m < p ∗ = p N N − p , the potential V (x) ∈ C (R N) and 0 < inf x ∈ R N V (x) < sup x ∈ R N V (x) < ∞. By Nehari manifold approach and the technique of splitting a minimizing sequence, we prove the existence of positive solutions for the above problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonexistence of solutions to the quasilinear Schrödinger equations: (1) − Δ p u + V (x) | u | p − 2 u − Δ p (| u | 2 α) | u | 2 α − 2 u = g (x , u) , x ∈ R N , where p-Laplacian operator Δ p u = d i v (| ∇ u | p − 2 ∇ u) , 1 < p < N and α > 1 / 2 is a parameter. Under suitable conditions on g (x , u) , the nonexistence of solutions to Equation (1) is investigated. [ABSTRACT FROM AUTHOR]

We consider the multiplicity of nontrivial solutions of the following quasilinear elliptic system -div(|x|(-ap)|∇u|(p-2)∇u) + f₁(x)|u|(p-2) u = (α/(α + β))g(x)|u| (α-2) u|v| (β) + λh₁(x)|u| (γ-2) u + l₁(x), -div(|x|(-ap) |∇v| (p-2)∇v) + f₂(x)|v| (p-2) v = (β/(α + β))g(x)|v|(β-2) v|u|(α) + μh 2(x)|v|(γ-2)v + l 2(x), u(x) > 0, v(x) > 0, x ∈ ℝ(N), where λ, μ > 0, 1 < p < N, 1 < γ < p < α + β < p* = Np/(N-pd), 0 ≤ a < (N - p)/p, a ≤ b < a + 1, d = a + 1 - b > 0. The functions f₁(x), f₂(x), g(x), h₁(x), h₂(x), l₁(x), and l₂(x) satisfy some suitable conditions. We will prove that the problem has at least two nontrivial solutions by using Mountain Pass Theorem and Ekeland's variational principle.

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem [formula: see text]. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.