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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 $$-(p_i- 1)|u_i'(x)|^{p_i-2} u_i''(x)=\lambda F_{u_i}(x, u_1,\dots, u_n) h_i(x, u_i'),\quad x\in (a,b),$$ $$u_i(a)= u_i(b)= 0,\quad\text{for }1\le i\le n,$$ where $p_i> 1$ for $1\le i\le n$, $\lambda$ is a positive parameter, $a,b\in\bbfR$ with $a< b$, $h_i:[a,b]\times \bbfR\to[0,+\infty)$ is a bounded and continuous function with $m_i:= \text{inf}_{(x,t)\in [a,b]\times \bbfR} h_i(x,t)> 0$ for $1\le i\le n$, $F: [a,b]\times\bbfR^n\to \bbfR$ is a function such that the mapping $(t_1,t_2,\dots, t_n)\to F(x, t_1,t_2,\dots, t_n)$ is in $C^1$ in $\bbfR^n$ for all $x\in [a,b]$, $F_{u_i}$ is continuous in $[a,b]\times \bbfR^n$ for $1\le i\le n$, where $F_{u_i}$ denotes the partial derivative of $F$ with respect to $u_i$, and $$\sup_{|(t_1,\dots, t_n)|\le M}|F_{u_i}(x, t_1,\dots, t_n)|\in L^1([a,b])$$ for all $M> 0$ and all $1\le i\le n$.

34B15Nonlinear boundary value problems for ODE
58E50Applications of variational methods in infinite-dimensional spaces
Full Text: DOI
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