## Topological Methods in Nonlinear Analysis

### On a singular semilinear elliptic problem: multiple solutions via critical point theory

#### Abstract

We study existence and multiplicity of solutions of a semilinear elliptic problem involving a singular term. Combining various techniques from critical point theory, under different sets of assumptions, we prove the existence of $k$ solutions ($k\in\mathbb N$) or infinitely many weak solutions.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 51, Number 2 (2018), 459-491.

Dates
First available in Project Euclid: 25 May 2018

https://projecteuclid.org/euclid.tmna/1527213956

Digital Object Identifier
doi:10.12775/TMNA.2018.002

Mathematical Reviews number (MathSciNet)
MR3829040

Zentralblatt MATH identifier
06928844

#### Citation

Faraci, Francesca; Smyrlis, George. On a singular semilinear elliptic problem: multiple solutions via critical point theory. Topol. Methods Nonlinear Anal. 51 (2018), no. 2, 459--491. doi:10.12775/TMNA.2018.002. https://projecteuclid.org/euclid.tmna/1527213956

#### References

• S. Agmon, The $L_p$ approach to the Dirichlet problem, Ann. Scuola Norm. Sup. Pisa 13 (1959), 405–448.
• A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids, C.R. Acad. Sci. Paris Sér. I Math. 305 (1987), 725–728.
• T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), 419–441.
• M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations \bf2 (1977), 193–222.
• R. Dhanya, E. Ko, R. Shivaji, A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl. 424 (2015), 598–612.
• F. Faraci and G. Smyrlis, Three solutions for a singular quasilinear elliptic problem, Proceedings of the Edinburgh Math. Soc. (to appear).
• F. Faraci and G. Smyrlis, Three solutions for a class of higher dimensional singular problems, Nonlinear Differ. Equ. Appl. \bf23 (2016), 14 pp.
• W. Fulks and J.S. Maybee, A singular non-linear equation, Osaka Math. J. \bf12 (1960), 1–19.
• J. Giacomoni and K. Saoudi, $W^{1,p}_0$ versus $C^1$ local minimizer for a singular and critical functional, J. Math. Anal. Appl. 363 (2010), 697–710.
• J. Giacomoni, I. Schindler and P. Takáč, Sobolev versus Hölder local minimizers nd existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Ser. V 6 (2007), 117–158.
• N. Hirano, C. Saccon and N. Shioji, Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations 245 (2008), 1997–2037.
• G.M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.
• A. Kristály, Detection of arbitrarily many solutions for perturbed elliptic problems involving oscillatory terms, J. Differential Equations \bf245 (2008), 3849–3868.
• D. Motreanu, V. Motreanu, N.S. Papageorgiou, A degree theoretic approach for multiple solutions of constant sign for nonlinear elliptic equations, Manuscripta Math. 124 (2007), 507–531.
• D. Motreanu, V. Motreanu, N.S. Papageorgiou, Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, 2014.
• N.S. Papageorgiou and G. Smyrlis, Nonlinear elliptic equations with singular reaction, Osaka J. Math. 53 (2016), 489–514.
• K. Perera and E.A.B. Silva, Existence and Multiplicity of positive solutions for singular quasilinear problems, J. Math. Anal. Appl. 323 (2006), 1238–1252.
• P. Pucci and J. Serrin, The Maximum Principle. Progress in Nonlinear Differential Equations and their Applications, Vol. 73, Birkhäuser Verlag, Basel, 2007.
• P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations 60 (1985), 142–149.
• G. Smyrlis, Nontrivial solutions for nonlinear problems with one sided resonance, Electron. J. Differential Equations 2012 (2012), 1–14.
• Y. Sun, S. Wu and Y. Long, Combined effects of singular and superlinear nonlinearities in some singular boundary value problems, J. Differential Equations 176 (2001), 511–531.
• J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), 191–202.
• L. Zhao, Y. He and P. Zhao, The existence of three positive solutions of a singular p-Laplacian problem, Nonlinear Anal. 74 (2011), 5745–5753.