## Taiwanese Journal of Mathematics

### Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum

Sarika Goyal

This article is in its final form and can be cited using the date of online publication and the DOI.

#### Abstract

In this article, we consider the following problem $\begin{cases} (-\Delta)^s u = \alpha u^+ - \beta u^{-} + f(u) + h &\textrm{in \Omega}, \\ u = 0 &\textrm{on \mathbb{R}^n \setminus \Omega}, \end{cases}$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain with Lipschitz boundary, $n \gt 2s$, $0 \lt s \lt 1$, $(\alpha,\beta) \in \mathbb{R}^2$, $f \colon \mathbb{R} \to \mathbb{R}$ is a bounded and continuous function and $h \in L^2(\Omega)$. We prove the existence results in two cases: first, the nonresonance case where $(\alpha,\beta)$ is not an element of the Fučik spectrum. Second, the resonance case where $(\alpha,\beta)$ is an element of the Fučik spectrum. Our existence results follows as an application of the saddle point theorem. It extends some results, well known for Laplace operator, to the nonlocal operator.

#### Article information

Source
Taiwanese J. Math., Advance publication (2020), 25 pages.

Dates
First available in Project Euclid: 4 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1583290813

Digital Object Identifier
doi:10.11650/tjm/200206

Subjects
Primary: 35A15: Variational methods 35B33: Critical exponents 35H39

#### Citation

Goyal, Sarika. Nonresonance and Resonance Problems for Nonlocal Elliptic Equations with Respect to the Fučik Spectrum. Taiwanese J. Math., advance publication, 4 March 2020. doi:10.11650/tjm/200206. https://projecteuclid.org/euclid.twjm/1583290813

#### References

• C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana 20, Springer, Bologna, 2016.
• L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), no. 2, 425–461.
• A. Castro, Hammerstein integral equations with indefinite kernel, Internat. J. Math. Math. Sci. 1 (1978), no. 2, 187–201.
• A. Castro and C. Chang, A variational characterization of the Fucik spectrum and applications, Rev. Colombiana Mat. 44 (2010), no. 1, 23–40.
• M. Cuesta, D. de Figueiredo and J.-P. Gossez, The beginning of the Fučik spectrum for the $p$-Laplacian, J. Differential Equations 159 (1999), no. 1, 212–238.
• E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573.
• P. Drábek and S. B. Robinson, On the Fredholm alternative for the Fučík spectrum, Abstr. Appl. Anal. 2010 (2010), Art. ID 125464, 20 pp.
• ––––, On the solvability of resonance problems with respect to the Fučik spectrum, J. Math. Anal. Appl. 418 (2014), no. 2, 884–905.
• M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 (2014), no. 2, 354–397.
• D. de Figueiredo and J.-P. Gossez, On the first curve of the Fučik spectrum of an elliptic operator, Differential Integral Equations 7 (1994), no. 5-6, 1285–1302.
• A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appli. Sci. 38 (2015), no. 16, 3551–3563.
• S. Goyal and K. Sreenadh, On the Fučik spectrum of non-local elliptic operators, NoDEA Nonlinear Differential equations Appl. 21 (2014), no. 4, 567–588.
• E. M. Landesman and A. C. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1969/1970), 609–623.
• E. Massa, On a variational characterization of the Fučík spectrum of the Laplacian and a superlinear Sturm-Liouville equation, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 3, 557–577.
• R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), no. 2, 887–898.
• ––––, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137.
• ––––, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat. 58 (2014), no. 1, 133–154.
• L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112.