## Topological Methods in Nonlinear Analysis

### Existence of multiple solutions for a quasilinear elliptic problem

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 50, Number 2 (2017), 531-551.

Dates
First available in Project Euclid: 11 October 2017

https://projecteuclid.org/euclid.tmna/1507687546

Digital Object Identifier
doi:10.12775/TMNA.2017.019

Mathematical Reviews number (MathSciNet)
MR3747027

Zentralblatt MATH identifier
06836832

#### Citation

Cossio, Jorge; Herrón, Sigifredo; Vélez, Carlos. Existence of multiple solutions for a quasilinear elliptic problem. Topol. Methods Nonlinear Anal. 50 (2017), no. 2, 531--551. doi:10.12775/TMNA.2017.019. https://projecteuclid.org/euclid.tmna/1507687546

#### 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 $p$-Laplacian 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 $p$-Laplacian, J. Differential Equations 92 (1991), 226–251.
• E. DiBenedetto, $\mathcal {C}^{1+\alpha}$ 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\rom: bifurcation from infinity, existence and multiplicity, Indiana Univ. Math. J. 53 (2004), No. 2, 433–482.
• P. Drábek, P. Krejčí and P. Takác, Nonlinear Differential Equations, Chapman, 1999.
• S. Fučík, J. Nečas, J. Souček and V. Souček, Spectral analysis of nonlinear operators, Lecture Notes in Mathematics, 346, Springer Verlag, 1973.
• J. García 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.