A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system. (English) Zbl 1244.34024
The authors establish the existence of at least three classical solutions for the Dirichlet quasilinear elliptic system
\[
-(p_i-1)|u_i'(x)|^{p_i-2}u_i''(x)=[\lambda F_{u_i}(x,u_1,\cdots,u_n)+\mu G_{u_i}(x,u_1,\cdots,u_n)]h_i(x,u_i') \text{ for } x\in (a,b),
\]
\[ u_i(a)=u_i(b) =0 \text{ for } i=1,\cdots,n. \] An example is also given for the applicability of the main result. The proof is based on a recent three critical points theorem of B. Ricceri [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, A, 3084–3089 (2009; Zbl 1214.47079)].
\[ u_i(a)=u_i(b) =0 \text{ for } i=1,\cdots,n. \] An example is also given for the applicability of the main result. The proof is based on a recent three critical points theorem of B. Ricceri [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 9, A, 3084–3089 (2009; Zbl 1214.47079)].
Reviewer: Zhiqing Han (Dalian)
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
Keywords:
three solutions; critical point; multiplicity results; Dirichlet problem with \(p\)-LaplacianCitations:
Zbl 1214.47079References:
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