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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. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and multiplicity of weak solutions to a class of quasilinear elliptic p⃗(x)$\vec{p}(x)$‐Kirchhoff type problems with weight and a nonlinear Robin boundary condition such as a+bK∑i=1N∫Ω1pi(x)∂u∂xipi(x)dx(−Δp⃗(x)u)+∑i=1NVi(x)upi(x)−2u=θ(x)um(x)−2u+f(x,u)inΩ,∑i=1N∂u∂xipi(x)−2∂u∂xiυi=η|u|q(x)−2uon∂Ω,\begin{equation*}\hskip7pc {\begin{cases} \displaystyle {\left(a+b K{\left(\sum _{i=1}^{N}\int \nolimits _ {\Omega } \frac{1}{p_{i}(x)} {\left|\frac{\partial u}{\partial x_{i}}\right|}^{p_{i}(x)} \mathrm{d}x\right)} \right)} \big (-\Delta _{\vec{p}(x)} u\big)+\sum _{i=1}^{N}V_{i}(x){\left|u\right|}^{p_{i}(x)-2}u\\[12pt] \quad =\theta (x){\left|u\right|}^{m(x)-2}u+f(x,u) \text{ in } \Omega , \\[2pt] \displaystyle \sum _{i=1}^{N}{\left| \frac{\partial u}{\partial x_{i}}\right|}^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\upsilon _{i} =\eta | u|^{q(x)-2}u \quad \text{on } \partial \Omega , \end{cases}}\hskip-7pc \end{equation*}where Ω is a smooth bounded domain. Under suitable conditions on the data, we show the existence and multiplicity of weak solutions by means of a variational approach in the framework of anisotropic Sobolev spaces with variable exponents. [ABSTRACT FROM AUTHOR]

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 paper studies a time discretization for a doubly nonlinear parabolic equation related to the p(x)-Laplacian by using Euler-forward scheme. We investigate existence, uniqueness, and stability questions and prove existence of the global compact attractor. [ABSTRACT FROM AUTHOR]

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f  in  Ω,  u=0  on  ∂Ω,}, where 00 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation. [ABSTRACT FROM AUTHOR]

We consider the Dirichlet initial boundary value problem ∂tum(x)-div⁡∇upx,t-2∇u=ax,tuq(x,t), where the exponents p(x,t)>1, q(x,t)>0, and m(x)>0 are given functions. We assume that a(x,t) is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if ess⁡sup⁡p(x,t)-10, in the case where a(x,t)

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]

This paper is concerned with the existence of time periodic solutions for parabolic equations associated with the singular p-Laplacian. The results are obtained by using pseudo-monotony, fixed point arguments, and an ε version of a standard inequality. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with a class of elliptic problems with both absorption terms and critical growth in the gradient. We suppose that the data belong to Lm() with m > n/2 and we prove the existence of bounded weak solutions via L1-estimates. A priori estimates and Stampacchia's L1-regularity are our main ingredient. [ABSTRACT FROM AUTHOR]

In this paper, we study existence, uniqueness and stability questions for the nonlinear parabolic equation: ∂u/∂t - Δpu + α(u) = f in ]0, T[xΩ, with Neumann-type boundary conditions and initial data in L¹. Our approach is based essentially on the time discretization technique by Euler forward scheme. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and boundedness of solutions of a doubly nonlinear parabolic system. We also obtain the existence of a global attractor and the regularity property for this attractor in [L∞(Ω)]² and ... [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and boundedness of solutions of a doubly nonlinear parabolic system. We also obtain the existence of a global attractor and the regularity property for this attractor in [L∞(Ω)]² and... [ABSTRACT FROM AUTHOR]

In this paper, we study a non-local initial boundary-value problem arising in Ohmic heating. By using a dynamical systems approach, some existence and uniqueness results are proved and the existence of a compact attractor is shown. [ABSTRACT FROM AUTHOR]

We study the doubly nonlinear parabolic equation ∂β(u)/∂t - Δpu + f(x, t, u) = 0 in Ω x R+, with Dirichlet boundary conditions and initial data. We investigate a time-discretization of the continuous problem by the Euler forward scheme. In addition to proving existence, uniqueness and stability questions, we study the long time behavior of the solution to the discrete problem. We prove the existence of a global attractor, and obtain its regularity under additional conditions. [ABSTRACT FROM AUTHOR]