Existence and regularity of solutions for a class of singular (p(x), q(x))-Laplacian systems

“…Papers that consider such problems by using the mentioned method are rare in the literature. Among such works we mention papers such as [2,3,24,34].…”

Section: Introductionmentioning

confidence: 99%

The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

Figueiredo

Razani

2021

Bound Value Probl

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In this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

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“…For example, nonlinear problems with the ( p , q )‐Laplace operator are considered in Motreanu et al 34,35 (systems of equations), Nastasi 36 (Riemann manifold), and Vetro 37 (mixed Dirichlet‐Neumann problems). For the existence and multiplicity of solutions of problems involving the ( p ( x ), q ( x ))‐Laplace operator, we refer to Alves and Moussaoui, 38 Papageorgiou and Vetro, 39 and Vetro and Vetro 40 . Also, we mention the results of Molica et al 41 (Neumann problems) and Papageorgiou et al 42 (double phase eigenvalue problem driven by the ( p , q )‐Laplacian plus an indefinite and unbounded potential, and Robin boundary condition).…”

Section: Introductionmentioning

confidence: 99%

On (p(x), q(x))‐Laplace equations in ℝN without Ambrosetti‐Rabinowitz condition

Nastasi

2021

Math Methods in App Sciences

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In the present work, we consider a (p(x), q(x))‐elliptic equation describing the behavior of a double‐phase anisotropic problem which has relevance in electrorheological fluid applications. The analysis leads to the existence of weak (nonnegative) solutions in the special case of potential terms with critical frequency and a superlinear reaction term. In order to prove the existence result, we combine critical point theory of mountain pass type with related topological and variational methods. Basically, the approach is variational, but we do not impose any Ambrosetti‐Rabinowitz type condition for the superlinearity of the reaction. More specifically, we apply the Euler‐Lagrange functional approach to the variational formulation of the above‐mentioned model problem. We note that we work in the whole space ℝN and so we have to consider non‐compact embeddings. This aspect constitutes an additional difficulty in our study.

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Existence and regularity of solutions for a class of singular (p(x), q(x))-Laplacian systems

Abstract: In this paper we study the existence of positive smooth solutions for a class of singular (p(x), q(x))-Laplacian systems by using sub and supersolution methods.2010 Mathematics Subject Classification. 35J75; 35J48; 35J92.

Cited by 18 publications

References 37 publications

On positive weak solutions for a class of weighted (p(.), q(.))−Laplacian systems

On positive weak solutions for a class of weighted (p(.), q(.))−Laplacian systems

The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

On (p(x), q(x))‐Laplace equations in ℝN without Ambrosetti‐Rabinowitz condition

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