Existence and Multiplicity Results for a Class of Coupled Quasilinear Elliptic Systems of Gradient Type
نویسندگان
چکیده
The aim of this paper is investigating the existence one or more weak solutions coupled quasilinear elliptic system gradient type \[ (P)\qquad \left\{ \begin{array}{ll} - {\rm div} (A(x, u)\vert\nabla u\vert^{p_1 -2} \nabla u) + \frac{1}{p_1}A_u (x, u\vert^{p_1} = G_u(x, u, v) &\hbox{ in $\Omega$,}\\[5pt] (B(x, v)\vert\nabla v\vert^{p_2 +\frac{1}{p_2}B_v(x, v\vert^{p_2} G_v\left(x, v\right) u v 0 on $\partial\Omega$,} \end{array} \right. \] where $\Omega \subset \mathbb{R}^N$ an open bounded domain, $p_1$, $p_2 > 1$ and $A(x,u)$, $B(x,v)$ are $\mathcal{C}^1$-Carath\'eodory functions \times \mathbb{R}$ with partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$, while $G_u(x,u,v)$, $G_v(x,u,v)$ given Carath\'eodory maps defined \mathbb{R}\times which a function $G(x,u,v)$. We prove that, even if coefficients make variational approach difficult, under suitable hypotheses functional $\cal{J}$, related to problem $(P)$, admits at least critical point ''right'' Banach space $X$. Moreover, $\cal{J}$ even, then $(P)$ has infinitely many solutions. proof, exploits interaction between two different norms, based version Cerami-Palais-Smale condition, ''good'' decomposition $X$ generalizations Ambrosetti-Rabinowitz Mountain Pass Theorems.
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ژورنال
عنوان ژورنال: Advanced Nonlinear Studies
سال: 2021
ISSN: ['1536-1365', '2169-0375']
DOI: https://doi.org/10.1515/ans-2021-2121