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Infinitely many solutions for a class of Dirichlet quasilinear elliptic systems. (English) Zbl 1253.34026

From the introduction: The aim of this paper is to investigate the existence of infinitely many classical solutions for the following Dirichlet quasilinear elliptic system

$-\left({p}_{i}-1\right){|{u}_{i}^{\text{'}}\left(x\right)|}^{{p}_{i}-2}{u}_{i}^{\text{'}\text{'}}\left(x\right)=\lambda {F}_{{u}_{i}}\left(x,{u}_{1},\cdots ,{u}_{n}\right){h}_{i}\left(x,{u}_{i}^{\text{'}}\right),\phantom{\rule{1.em}{0ex}}x\in \left(a,b\right),$
${u}_{i}\left(a\right)={u}_{i}\left(b\right)=0,\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}1\le i\le n,$

where ${p}_{i}>1$ for $1\le i\le n$, $\lambda$ is a positive parameter, $a,b\in ℝ$ with $a, ${h}_{i}:\left[a,b\right]×ℝ\to \left[0,+\infty \right)$ is a bounded and continuous function with ${m}_{i}:={\text{inf}}_{\left(x,t\right)\in \left[a,b\right]×ℝ}{h}_{i}\left(x,t\right)>0$ for $1\le i\le n$, $F:\left[a,b\right]×{ℝ}^{n}\to ℝ$ is a function such that the mapping $\left({t}_{1},{t}_{2},\cdots ,{t}_{n}\right)\to F\left(x,{t}_{1},{t}_{2},\cdots ,{t}_{n}\right)$ is in ${C}^{1}$ in ${ℝ}^{n}$ for all $x\in \left[a,b\right]$, ${F}_{{u}_{i}}$ is continuous in $\left[a,b\right]×{ℝ}^{n}$ for $1\le i\le n$, where ${F}_{{u}_{i}}$ denotes the partial derivative of $F$ with respect to ${u}_{i}$, and

$\underset{|\left({t}_{1},\cdots ,{t}_{n}\right)|\le M}{sup}|{F}_{{u}_{i}}\left(x,{t}_{1},\cdots ,{t}_{n}\right)|\in {L}^{1}\left(\left[a,b\right]\right)$

for all $M>0$ and all $1\le i\le n$.

##### MSC:
 34B15 Nonlinear boundary value problems for ODE 58E50 Applications of variational methods in infinite-dimensional spaces
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