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Front. Math. China 2016, 11(6): 1613–1623DOI 10.1007/s11464-016-0584-1Existence of saddle solutions of a nonlinearelliptic equation involving p -Laplacian inmore even-dimensional spacesHuahui YAN, Zhuoran DUCollege of Mathematics and Econometrics, Hunan University, Changsha 410082, Chinac ? Higher Education Press and Springer-Verlag Berlin Heidelberg 2016Abstract We show that there exist saddle solutions of the nonlinear ellipticequation involving the p-Laplacian, p > 2, −Δ p u = f(u) in R 2m for alldimensions satisfying 2m ? p, by using sub-supersolution method. Theexistence of saddle solutions of the above problem was known only in dimensions2m ? 2p.Keywords p-Laplacian, saddle solutions, sub-supersolution methodMSC 35J60, 35J70, 34C371 IntroductionIn this paper, we study the existence of saddle solutions of the nonlinear ellipticequation involving the p-Laplacian operator−Δ p u := −div(|∇u| p−2 ∇u) = f(u) in R 2m , (1)where 2m ? p > 2. Nonlinearity f(u) satisf i es several conditions (7)–(10) asstated at the end of this section.Saddle solutions were f i rst studied by Dang et al. [6] for equation −Δu =f(u) in R 2 , with f odd, bistable, and f(u)/u decreasing for u ∈ (0,1). Theyproved the existence and uniqueness of a saddle solution. They also establishedmonotonicity properties and the asymptotic behavior of the saddle solution.Schatzman [11] studied in detail its instability, which was already indicated ina partial result of [6]. The nondegeneracy of the saddle solution was proved byKowalczyk and Liu [10]. Alama et al. [1] studied vector-valued saddle solutionsin R 2 . The article [2] concerns scalar saddle type solutions in R 2 changing signon more nodal lines than x 1 = ±x 2 . Recently, Cabr´ e and Terra [4] proved theReceived April 9, 2014; accepted August 22, 2016Corresponding author: Zhuoran DU, E-mail: duzr@hnu.edu.cn