Existence of minimizers for some quasilinear elliptic problems
Abstract: The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear elliptic problem − div(a(x, u, ∇u)) + At(x, u, ∇u) = f (x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ R N is an open bounded domain and A(x, t, ξ), f (x, t) are given real functions, with At = ∂A ∂t , a = ∇ ξ A. We prove that, even if A(x, t, ξ) makes the variational approach more difficult, the functional associated to such a problem is bounded from below and attains its infimum when the growth of the nonlinear … Show more
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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 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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