Abstract
Consider the quasilinear elliptic equation
where \(\Omega \subset \mathbb R^N\) (\(N\ge 2\)) is a bounded domain with smooth boundary and \(\lambda >0\) is a parameter. For the quasilinear term, we assume that \(a_{ij}=a_{ji}\) and \(a_{ij}(x,s)\)’s growth is like \((1+s^{2})\delta _{ij}\). The nonlinearity of power growth \(f(x,s)=|s|^{r-2}s\) with \(2<r<4\) acts as a typical example of the nonlinear term f, a case in which less results are known compared with the cases \(1<r<2\) and \(4<r<4N/(N-2)^+\). We show the structure of solutions depends keenly upon the parameter \(\lambda \). More precisely, while such an equation has no nontrivial solution for \(\lambda \) small, we prove that both the number of solutions with positive energies and the number of solutions with negative energies tend to infinity as \(\lambda \rightarrow +\infty \). Nodal properties are determined for six solutions among all of them.
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Acknowledgements
The authors would like to thank the referee for their useful comments. Y. Jing is supported by National Natural Science Foundation of China (No. 11271265). Z. Liu is supported by National Natural Science Foundation of China (Nos. 11271265, 11331010) and Beijing Center for Mathematics and Information Interdisciplinary Sciences. Z.-Q. Wang is supported by National Natural Science Foundation of China (No. 11271201) and Beijing Center for Mathematics and Information Interdisciplinary Sciences.
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Jing, Y., Liu, Z. & Wang, ZQ. Multiple solutions of a parameter-dependent quasilinear elliptic equation. Calc. Var. 55, 150 (2016). https://doi.org/10.1007/s00526-016-1067-7
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DOI: https://doi.org/10.1007/s00526-016-1067-7