Nontrivial solutions for a class of gradient-type quasilinear elliptic systems
Abstract: The aim of this paper is investigating the existence of weak bounded solutions of the gradient-type quasilinear elliptic system (P )with m ≥ 2 and u = (u1, . . . , um), where Ω ⊂ R N is an open bounded domain and some functions Ai : Ω × R × R N → R, i ∈ {1, . . . , m}, and G : Ω × R m → R exist such that ai(x, t, ξ) = ∇ ξ Ai(x, t, ξ), Ai,t(x, t, ξ) = ∂A i ∂t (x, t, ξ) and Gi(x, u) = ∂G ∂u i (x, u). We prove that, under suitable hypotheses, the functional J related to problem (P ) is C 1 on a "good" Banach spac… Show more
Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Cited by 1 publication
References 17 publications
Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth
In this paper, we consider the following coupled gradient-type quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{*{20}l} - {\text{div}} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &{}{\hbox { in }}\Omega ,\\ - {\text{div}} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left( x, u, v\right) &{}{\hbox { in }}\Omega ,\\ u = v = 0 &{}{\hbox { on }}\partial \Omega , \end{array} \right. \end{aligned}$$ - div ( a ( x , u , ∇ u ) ) + A t ( x , u , ∇ u ) = G u ( x , u , v ) in Ω , - div ( b ( x , v , ∇ v ) ) + B t ( x , v , ∇ v ) = G v x , u , v in Ω , u = v = 0 on ∂ Ω , where $$\Omega$$ Ω is an open bounded domain in $${\mathbb {R}}^N$$ R N , $$N\ge 2$$ N ≥ 2 . We suppose that some $$\mathcal {C}^{1}$$ C 1 –Carathéodory functions $$A, B:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ A , B : Ω × R × R N → R exist such that $$a(x,t,\xi ) = \nabla _{\xi } A(x,t,\xi )$$ a ( x , t , ξ ) = ∇ ξ A ( x , t , ξ ) , $$A_t(x,t,\xi ) = \frac{\partial A}{\partial t} (x,t,\xi )$$ A t ( x , t , ξ ) = ∂ A ∂ t ( x , t , ξ ) , $$b(x,t,\xi ) = \nabla _{\xi } B(x,t,\xi )$$ b ( x , t , ξ ) = ∇ ξ B ( x , t , ξ ) , $$B_t(x,t,\xi ) =\frac{\partial B}{\partial t}(x,t,\xi )$$ B t ( x , t , ξ ) = ∂ B ∂ t ( x , t , ξ ) , and that $$G_u(x, u, v)$$ G u ( x , u , v ) , $$G_v(x, u, v)$$ G v ( x , u , v ) are the partial derivatives of a $$\mathcal {C}^{1}$$ C 1 –Carathéodory nonlinearity $$G:\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ G : Ω × R × R → R . Roughly speaking, we assume that $$A(x,t,\xi )$$ A ( x , t , ξ ) grows at least as $$(1+|t|^{s_1p_1})|\xi |^{p_1}$$ ( 1 + | t | s 1 p 1 ) | ξ | p 1 , $$p_1 > 1$$ p 1 > 1 , $$s_1 \ge 0$$ s 1 ≥ 0 , while $$B(x,t,\xi )$$ B ( x , t , ξ ) grows as $$(1+|t|^{s_2p_2})|\xi |^{p_2}$$ ( 1 + | t | s 2 p 2 ) | ξ | p 2 , $$p_2 > 1$$ p 2 > 1 , $$s_2 \ge 0$$ s 2 ≥ 0 , and that G(x, u, v) can also have a supercritical growth related to $$s_1$$ s 1 and $$s_2$$ s 2 . Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.
Multiple solutions for coupled gradient-type quasilinear elliptic systems with supercritical growth
In this paper, we consider the following coupled gradient-type quasilinear elliptic system $$\begin{aligned} \left\{ \begin{array}{*{20}l} - {\text{div}} ( a(x, u, \nabla u) ) + A_t (x, u, \nabla u) = G_u(x, u, v) &{}{\hbox { in }}\Omega ,\\ - {\text{div}} ( b(x, v, \nabla v) ) + B_t(x, v, \nabla v) = G_v\left( x, u, v\right) &{}{\hbox { in }}\Omega ,\\ u = v = 0 &{}{\hbox { on }}\partial \Omega , \end{array} \right. \end{aligned}$$ - div ( a ( x , u , ∇ u ) ) + A t ( x , u , ∇ u ) = G u ( x , u , v ) in Ω , - div ( b ( x , v , ∇ v ) ) + B t ( x , v , ∇ v ) = G v x , u , v in Ω , u = v = 0 on ∂ Ω , where $$\Omega$$ Ω is an open bounded domain in $${\mathbb {R}}^N$$ R N , $$N\ge 2$$ N ≥ 2 . We suppose that some $$\mathcal {C}^{1}$$ C 1 –Carathéodory functions $$A, B:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ A , B : Ω × R × R N → R exist such that $$a(x,t,\xi ) = \nabla _{\xi } A(x,t,\xi )$$ a ( x , t , ξ ) = ∇ ξ A ( x , t , ξ ) , $$A_t(x,t,\xi ) = \frac{\partial A}{\partial t} (x,t,\xi )$$ A t ( x , t , ξ ) = ∂ A ∂ t ( x , t , ξ ) , $$b(x,t,\xi ) = \nabla _{\xi } B(x,t,\xi )$$ b ( x , t , ξ ) = ∇ ξ B ( x , t , ξ ) , $$B_t(x,t,\xi ) =\frac{\partial B}{\partial t}(x,t,\xi )$$ B t ( x , t , ξ ) = ∂ B ∂ t ( x , t , ξ ) , and that $$G_u(x, u, v)$$ G u ( x , u , v ) , $$G_v(x, u, v)$$ G v ( x , u , v ) are the partial derivatives of a $$\mathcal {C}^{1}$$ C 1 –Carathéodory nonlinearity $$G:\Omega \times {\mathbb {R}}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ G : Ω × R × R → R . Roughly speaking, we assume that $$A(x,t,\xi )$$ A ( x , t , ξ ) grows at least as $$(1+|t|^{s_1p_1})|\xi |^{p_1}$$ ( 1 + | t | s 1 p 1 ) | ξ | p 1 , $$p_1 > 1$$ p 1 > 1 , $$s_1 \ge 0$$ s 1 ≥ 0 , while $$B(x,t,\xi )$$ B ( x , t , ξ ) grows as $$(1+|t|^{s_2p_2})|\xi |^{p_2}$$ ( 1 + | t | s 2 p 2 ) | ξ | p 2 , $$p_2 > 1$$ p 2 > 1 , $$s_2 \ge 0$$ s 2 ≥ 0 , and that G(x, u, v) can also have a supercritical growth related to $$s_1$$ s 1 and $$s_2$$ s 2 . Since the coefficients depend on the solution and its gradient themselves, the study of the interaction of two different norms in a suitable Banach space is needed. In spite of these difficulties, a variational approach is used to show that the system admits a nontrivial weak bounded solution and, under hypotheses of symmetry, infinitely many ones.
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
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.
