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In this paper, we prove the existence of weak solutions to Neumann boundary value problems for a nonlinear p(x)-elliptic equation of the form -div a(x,u,∇u)=b(x)| u |p(x)-2u+λH(x,u, ∇u). We established the existence result by using the topological degree introduced by Berkovits.

In the present paper, we will study the existence of at least one weak solution for the nonlinear parabolic initial boundary value problem associated with the following equation of Kirchoff type ∂ ∂ − M ( ∫ Ω ( B ( x , t , ∇ ⁡ u ) + 1 | ∇ ⁡ u | ) d x ) div ( b ( x , t , ∇ ⁡ u ) + | ∇ ⁡ u | − 2 ∇ ⁡ u ) = f ( x , t ) − g ( x , t , u , ∇ ⁡ u ) . By using the Topological degree theory for operators of the type + + , where is a maximal monotone map, is bounded demicontinuous map of class ( S + ) and be compact and belongs to Γ σ τ (i.e there exist τ , σ ≥ 0 such that ∥ x ∥ ≤ τ ∥ x ∥ + σ ) . Our focus of the study is centered on this problem in space L ( 0 , T , W 1 , ( Ω ) ) , where ≥ 2 and Ω is a bounded open domain in ℝ N .

In this paper we study a Neumann boundary value problem of a new p(x)-Kirchhoff type problems driven by p(x)-Laplacian-like operators. Using the theory of variable exponent Sobolev spaces and the method of the topological degree for a class of demicontinuous operators of generalized (S ) type,weprove theexistenceofaweak solutionsof this problem. Our results are a natural generalisation of some existing ones in the context of p(x)-Kirchhoff type problems.

In this paper, we will use the topological degree, introduced by Berkovits, to prove existence of weak solutions to a Neumann boundary value problems for the following nonlinear elliptic equation where Ω is a bounded smooth domain of 𝕉

In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form where the right hand side belongs to (Ω). The operator is a Leray-Lions anisotropic operator and ∈ (ℝ,ℝ).

This work discusses the existence of weak solutions for a system of Kirchhoff-type involving variable exponent (α₁(m), α₂(m))-Laplacian operators and under the Dirichlet boundary conditions. Under appropriate hypotheses on the nonlinear terms and the Kirchhoff functions, the existence of weak solutions is obtained on the spaces W 0 1 , α 1 ( m ) ( D ) × W 0 1 , α 2 ( m ) ( D ) . The proof of the main result is based on a topological degree argument for a class of demicontinuous operators of (S₊)-type.

Our purpose is to establish the existence of weak solutions to Neumann boundary value problem for equations involving the p ( x )-Laplacian-like operator and the p ( x )-Laplacian operator. The existence proof is based on the theory of the variable exponent Sobolev spaces and the topological degree theory. Our result extend and generalize several corresponding results from the existing literature.

In this paper, we study the existence of weak solutions for a nonlocal elliptic system involving the ( p ( x ),  q ( x ))-Kirchhoff–Laplacian operators with Dirichlet boundary conditions, in the case of a reaction term depending also on the gradient (convection). Using a topological degree for a class of demicontinuous operators of generalized ( S + ) type, we obtain the existence result of weak solutions of the considered problem in the framework of Sobolev space with variable exponent. Our results extend and generalize some recent works in the existing literature.

In this paper, we consider the 3-D generalized micropolar fluid system in critical Fourier-Besov-Morrey spaces with variable exponent. Using the Littlewood-Paley theory and Banach fixed point theorem we establish the global existence result with the small initial data belonging to F N ˙ p ( ⋅ ) , h ( ⋅ ) , q 4 − 2 α − 3 p ( ⋅ ) ( ℝ 3 ) .

This research establishes the existence of weak solution for a Dirichlet boundary value problem involving the p ( x )-Laplacian-like operator depending on three real parameters, originated from a capillary phenomena, of the following form: - Δ p ( x ) l u + δ | u | α ( x ) - 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) i n Ω , u = 0 o n ∂ Ω , where Δ p ( x ) l is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , δ , μ , and λ are three real parameters, and p ( · ) , α ( · ) ∈ C + ( Ω ¯ ) and g ,  f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable-exponent Sobolev spaces, we establish the existence of a weak solution for the above problem.