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Cited by 5 publications
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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.
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.
In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation $$\begin{aligned} - \textrm{div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) + V(x) {\vert u \vert }^{p-2} u= g(x,u) \quad \quad \hbox { in }{{\mathbb {R}}}^{N} \end{aligned}$$ - div ( a ( x , u , ∇ u ) ) + A t ( x , u , ∇ u ) + V ( x ) | u | p - 2 u = g ( x , u ) in R N with $$p>1$$ p > 1 , $$N\ge 2\ $$ N ≥ 2 and $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ V : R N → R suitable measurable positive function. Here, we suppose $$A: {\mathbb {R}}^N \times {\mathbb {R}}\times {\mathbb {R}}^N \rightarrow {\mathbb {R}}$$ A : R N × R × R N → R is a given $${C}^{1}$$ C 1 -Carathéodory function which grows as $$|\xi |^p$$ | ξ | p , with $$A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$$ A t ( x , t , ξ ) = ∂ A ∂ t ( x , t , ξ ) , $$a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )$$ a ( x , t , ξ ) = ∇ ξ A ( x , t , ξ ) , $$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$ V : R N → R is a suitable measurable function and $$g:{\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ g : R N × R → R is a given Carathéodory function which grows as $$|\xi |^q$$ | ξ | q with $$1<q<p$$ 1 < q < p . Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on $$A(x,t,\xi ), V(x)$$ A ( x , t , ξ ) , V ( x ) and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.
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