Multiplicity results for some nonlinear elliptic problems with asymptotically $${{\varvec{p}}}$$ p -linear terms
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Cited by 11 publications
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<p style='text-indent:20px;'>The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb R^N $\end{document}</tex-math></inline-formula> is an open bounded domain, <inline-formula><tex-math id="M2">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ A(x,t,\xi) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ B(x,t, {\xi}) $\end{document}</tex-math></inline-formula> are <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{C}^1 $\end{document}</tex-math></inline-formula>–Carathéodory functions on <inline-formula><tex-math id="M6">\begin{document}$ \Omega \times \mathbb R \times { \mathbb R}^{N} $\end{document}</tex-math></inline-formula> with partial derivatives <inline-formula><tex-math id="M7">\begin{document}$ A_t = \frac{\partial A}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ a = {\nabla}_{\xi}A $\end{document}</tex-math></inline-formula>, respectively <inline-formula><tex-math id="M9">\begin{document}$ B_t = \frac{\partial B}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ b = {\nabla}_{{\xi}}B $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M11">\begin{document}$ g_1(x,t,s) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ g_2(x,t,s) $\end{document}</tex-math></inline-formula> are given Carathéodory maps defined on <inline-formula><tex-math id="M13">\begin{document}$ \Omega \times \mathbb R\times \mathbb R $\end{document}</tex-math></inline-formula> which are partial derivatives with respect to <inline-formula><tex-math id="M14">\begin{document}$ t $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ s $\end{document}</tex-math></inline-formula> of a function <inline-formula><tex-math id="M16">\begin{document}$ G(x,t,s) $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We prove that, even if the general form of the terms <inline-formula><tex-math id="M17">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$ B $\end{document}</tex-math></inline-formula> makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space <inline-formula><tex-math id="M19">\begin{document}$ X $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem.</p>
<p style='text-indent:20px;'>The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb R^N $\end{document}</tex-math></inline-formula> is an open bounded domain, <inline-formula><tex-math id="M2">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ A(x,t,\xi) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ B(x,t, {\xi}) $\end{document}</tex-math></inline-formula> are <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{C}^1 $\end{document}</tex-math></inline-formula>–Carathéodory functions on <inline-formula><tex-math id="M6">\begin{document}$ \Omega \times \mathbb R \times { \mathbb R}^{N} $\end{document}</tex-math></inline-formula> with partial derivatives <inline-formula><tex-math id="M7">\begin{document}$ A_t = \frac{\partial A}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ a = {\nabla}_{\xi}A $\end{document}</tex-math></inline-formula>, respectively <inline-formula><tex-math id="M9">\begin{document}$ B_t = \frac{\partial B}{\partial t} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ b = {\nabla}_{{\xi}}B $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M11">\begin{document}$ g_1(x,t,s) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ g_2(x,t,s) $\end{document}</tex-math></inline-formula> are given Carathéodory maps defined on <inline-formula><tex-math id="M13">\begin{document}$ \Omega \times \mathbb R\times \mathbb R $\end{document}</tex-math></inline-formula> which are partial derivatives with respect to <inline-formula><tex-math id="M14">\begin{document}$ t $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ s $\end{document}</tex-math></inline-formula> of a function <inline-formula><tex-math id="M16">\begin{document}$ G(x,t,s) $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>We prove that, even if the general form of the terms <inline-formula><tex-math id="M17">\begin{document}$ A $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$ B $\end{document}</tex-math></inline-formula> makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space <inline-formula><tex-math id="M19">\begin{document}$ X $\end{document}</tex-math></inline-formula>.</p><p style='text-indent:20px;'>The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem.</p>
In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem −div(a(x, u, ∇u)) + At(x, u, ∇u) = f (x, u) in Ω, u ≥ 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ R N is an open bounded domain, N ≥ 3, and A(x, t, ξ), f (x, t) are given functions, with At = ∂A ∂t , a = ∇ ξ A. To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami-Palais-Smale condition. Furthermore, if A(x, t, ξ) grows fast enough with respect to t, then the nonlinear term related to f (x, t) may have also a 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.
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