Existence of multiple solutions for a quasilinear elliptic problem

Jorge Cossio, Sigifredo Herrón, Carlos Vélez

DOI: http://dx.doi.org/10.12775/TMNA.2017.019

Abstract


In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the $p$-derivative at zero and the $p$-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the $p$-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from zero to prove the existence of unbounded branches of positive solutions (resp. of negative solutions). We show the existence of multiple solutions and we provide qualitative properties of these solutions.

Keywords


Quasilinear elliptic equations; bifurcation theory; multiplicity of solutions

Full Text:

PREVIEW FULL TEXT

References


A. Ambrosetti, J. Garcia Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219–242.

A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (2), 1980, 411–422.

A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics 104, Cambridge University Press, 2007.

J. Cossio and S. Herrón, Existence of radial solutions for an asymptotically linear pLaplacian problem, J. Math. Anal. Appl. 345 (2008), 583–592.

J. Cossio, S. Herrón and C. Vélez, Multiple solutions for nonlinear Dirichlet problems via bifurcation and additional results, J. Math. Anal. Appl. 399 (2013), 166–179.

J. Cossio, S. Herrón and C. Vélez, Infinitely many radial solutions for a p-Laplacian problem p-superlinear at the origen, J. Math. Anal. Appl. 376 (2011), 741–749.

M. Cuesta Leon, Existence results for quasilinear problems via ordered sub and supersolutions, (English, French summary) Ann. Fac. Sci. Toulouse Math. (6) 6 (1997), No. 4, 591–608.

L.M. Del Pezzo and A. Quaas, Global bifurcation for fractional p-Laplacian and an application, arXiv:1412.4722v2 (2016).

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the pLaplacian, J. Differential Equations 92 (1991), 226–251.

E. DiBenedetto, C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), No. 8, 827–850.

P. Drábek, Asymptotic bifurcation problems for quasilinear equations, existence and multiplicity results, Topol. Methods Nonlinear Anal. 25 (2005), No. 1, 183–194.

P. Drábek, P. Girg, P. Takác and M. Ulm, The Fredholm alternative for the p-Laplacian: bifurcation from infinity, existence and multiplicity, Indiana Univ. Math. J. 53 (2004), No. 2, 433–482.

P. Drábek, P. Krejc and P. Takác, Nonlinear Differential Equations, Chapman, 1999.

S. Fučik, J. Nekas, J. Soucek and K. Soucek, Spectral analysis of nonlinear operators, Lecture Notes in Mathematics, 346, Springer Verlag, 1973.

J. Garcı́a Melián and J. Sabina de Lis, Uniqueness to quasilinear problems for the p-Laplacian in radially symmetric domains, Nonlinear Anal. 43 (2001), 803–835.

L. Gasinski and N. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications Vol. 9, Chapman & Hall, 2006.

A. Lê, On the local Holder continuity of the inverse of the p-Laplace operator, Proc. Amer. Math. Soc. 135 (2007), No. 11, 3353–3560.

G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), No. 11, 1203–1219.

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis 7, 487-513 (1971).

P.H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202.

P.H. Rabinowitz, Global aspects of bifurcation, Topological Methods in Bifurcation Theory, Sem. Math. Sup. 91, Univ. Montreal, Montreal, 1985, pp. 63–112.

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.

J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism