Resonant Anisotropic (p,q)-Equations
Abstract: We consider an anisotropic Dirichlet problem which is driven by the (p(z),q(z))-Laplacian (that is, the sum of a p(z)-Laplacian and a q(z)-Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x±∞ can be resonant with respect to the principal eigenvalue of (−Δp(z),W01,p(z)(Ω)). First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution … Show more
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 5 publications
References 40 publications
In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.
In this paper, we consider local Dirichlet problems driven by the (r(u),s(u))-Laplacian operator in the principal part. We prove the existence of nontrivial weak solutions in the case where the variable exponents r,s are real continuous functions and we have dependence on the solution u. The main contributions of this article are obtained in respect of: (i) Carathéodory nonlinearity satisfying standard regularity and polynomial growth assumptions, where in this case, we use geometrical and compactness conditions to establish the existence of the solution to a regularized problem via variational methods and the critical point theory; and (ii) Sobolev nonlinearity, somehow related to the space structure. In this case, we use a priori estimates and asymptotic analysis of regularized auxiliary problems to establish the existence and uniqueness theorems via a fixed-point argument.
We consider the Dirichlet boundary value problem for equations involving the $(p(z),q(z))$ ( p ( z ) , q ( z ) ) -Laplacian operator in the principal part on an open bounded domain $\Omega \subset \mathbb{R}^{n}$ Ω ⊂ R n . Here, the $p(z)$ p ( z ) -Laplacian is weighted by a function $a \in L^{\infty}(\Omega )_{+}$ a ∈ L ∞ ( Ω ) + , and the nonlinearity in the reaction term is allowed to depend on the solution without imposing the Ambrosetti–Rabinowitz condition. The proof of the existence of solution to our problem is based on a mountain pass critical point approach with the Cerami condition at level c.
Existence of solutions for (p(y),q(y))-Laplacian elliptic problem on an exterior domain
In this paper, we study the following elliptic problem involving the ( p ( y ) , q ( y ) {p(y),q(y)} )-Laplacian operator: { - div ( a ( y ) | ∇ v | p ( y ) - 2 ∇ v ) + b ( y ) | v | p ( y ) - 2 v - div ( | ∇ v | q ( y ) - 2 ∇ v ) = g ( y , v ) , y ∈ Ω , v = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle{}{-}\operatorname{div}(a(y)|\nabla v|^{p(% y)-2}\nabla v)+b(y)|v|^{p(y)-2}v-\operatorname{div}(|\nabla v|^{q(y)-2}\nabla v% )&\displaystyle=g(y,v),&&\displaystyle y\in\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. with Dirichlet boundary condition in an exterior domain Ω ( ⊂ ℝ n ) {(\subset\mathbb{R}^{n})} with smooth boundary, where 1 < q ( y ) < p ( y ) < n 1<q(y)<p(y)<n . We prove the existence of solutions in W 0 1 , p ( y ) ( Ω ) {W^{1,p(y)}_{0}(\Omega)} for the superlinear case by using the Mountain Pass Theorem.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
Contact Info
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Product
Resources
This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
