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In this paper, we study the solvability to the left of the positive infimum of all eigenvalues for some non-resonant quasilinear elliptic problems involving variable exponents. We first prove the existence of at least a weak solution for some non-variational systems by using a surjectivity result for pseudomonotone operators. Furthermore, under additional conditions, we show that the solution is unique and provide examples. Second, we deal with non-resonant gradient-type systems and obtain existence by using a variational approach.

In this paper, we consider the quasilinear elliptic problem with potential (P) − div (φ (x , | ∇ u |) ∇ u) + V (x) | u | q (x) − 2 u = f (x , u) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N ( N ≥ 2), V is a given function in a generalized Lebesgue space L s (x) (Ω) , and f (x , u) is a Carathéodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for (P) in the framework of Musielak–Sobolev spaces. The main difficulty here is that the nonlinearity f (x , u) considered does not satisfy the well-known Ambrosetti–Rabinowitz condition. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of at least three weak solutions for a parametric double eigenvalue quasi-linear elliptic p x , q x -Kirchhoff-type potential system. Our approach is based on a variational method, and a three critical point theorem is obtained by Bonano and Marano. [ABSTRACT FROM AUTHOR]

This article presents suffcient conditions for the existence of solutions of the anisotropic quasilinear elliptic equation with variable exponent and nonlinear Robin boundary conditions, -ΣNi=1φ/φxi (|φu/φxi)NΣi=1|u|pi(x)-2u + λ|u|m(x)-2u = γg(x,u) in Ω NΣi=1|φu/φxi|pi(x)-2 φu/φxi vi = μ|u|q(x-2)u on ∂Ω Under appropriate assumptions on the data, we prove some existence and multiplicity results. The methods are based on Mountain Pass and Fountain theorems. [ABSTRACT FROM AUTHOR]