Existence of saddle solutions of a nonlinear elliptic equation involving p-Laplacian in more even-dimensional spaces

Abstract

We show that there exist saddle solutions of the nonlinear elliptic equation involving the p-Laplacian, p>2, -{{\displaystyle \Delta }}_{p}u=f(u) in {ℝ}^{\mathrm{2}m} for all dimensions satisfying 2m≥p, by using sub-supersolution method. The existence of saddle solutions of the above problem was known only in dimensions 2m≥2p.

Keywords

p-Laplacian / saddle solutions / sub-supersolution method

References

[1] Alama S, Bronsard L, Gui C. Stationary layered solutions in R²for an Allen-Cahn system with multiple well potential. Calc Var Partial Differential Equations, 1997, 5: 359–390

[2] Alessio F, Calamai A, Montecchiari P. Saddle-type solutions for a class of semilinear elliptic equations. Adv Differential Equations, 2007, 12: 361–380

[3] Cabré X. Uniqueness and stability of saddle-shaped solutions to the Allen-Cahn equation. J Math Pure Appl, 2012, 98(3): 239–256

[4] Cabré X, Terra J. Saddle solutions of bistable diffusion equations in all of ℝ2m.J Eur Math Soc (JEMS), 2009, 11(4): 819–843

[5] Cabré X, Terra J. Qualitative properties of saddle-shaped solutions to bistable diffusion equations. Comm Partial Differential Equations, 2010, 35: 1923–1957

[6] Dang H, Fife P C, Peletier L. A, Saddle solutions of the bistable diffusion equation. Z Angew Math Phys, 1992, 43: 984–998

[7] Dibenedetto E. C^1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal, 1983, 7(8): 827–850

[8] Drábek P, Kerjci P, Takac P. Nonlinear Differential Equations. London: Chapman and Hall/CRC, 1999

[9] Du Z, Zhou Z, Lai B. Saddle solutions of nonlinear elliptic equations involving the p-Laplacian. NoDEA Nonlinear Differential Equations Appl, 2011, 18: 101–114

[10] Kowalczyk M, Liu Y. Nondegeneracy of the saddle solution of the Allen-Cahn equation. Proc Amer Math Soc, 2011, 139(12): 4319–4329

[11] Schatzman M. On the stability of the saddle solution of Allen-Cahn’s equation. Proc Roy Soc Edinburgh Sect A, 1995, 125: 1241–1275

[12] Sciunzi B, Valdinoci E. Mean curvature properties for p-Laplace phase transitions. J Eur Math Soc (JEMS), 2005, 7(3): 319–359

[13] Valdinoci E, Sciunzi B, Savin O. Flat Level Set Regularity of p-Laplace Phase Transitions. Mem Amer Math Soc, Vol 182, No 858. Providence: Amer Math Soc, 2006

[14] Vazquez J L. A strong maximum principle for some quasilinear elliptic equations. Appl Math Optim, 1984, 12(1): 191–202

[15] Yan H, Du Z. Maximal saddle solution of a nonlinear elliptic equation involving the p-Laplacian. Proc Indian Acad Sci Math Sci, 2014, 124(1): 57–65

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg