The multiplicity of solutions for the critical problem involving the fracional p-Laplacian operator

Resumo

This paper deals with the existence of multiple solutions

for the following critical fractional $p$-Laplacian problem

\begin{equation*}

\left\{

\begin{array}{l}

\mathbf{(-}\Delta \mathbf{)}_{p}^{s}u(x)=\lambda \left\vert u\right\vert

^{p-2}u+f(x,u)+\mu g(x,u)\ \text{in }\Omega ,u>0, \\

\\

u=0\text{ on}\ \mathbb{R}^{n}\setminus \Omega ,%

\end{array}%

\right.

\end{equation*}%

where $p>1$, $s\in (0,1)$, $\Omega \subset \mathbb{R}^{n}(n>ps),$ be a bounded smooth domain, $\lambda $, $\mu $ are positive parameters and the functions $f,g:\overline{%

\Omega }\times \lbrack 0,\infty )\longrightarrow [0,\infty),$ are continuous and differentiable with respect to the second variable. Our main tools are based on variational methods combined with a classical concentration

compacteness method.

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Referências

J. G. Azorero, I. P. Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Am. Math. Soc., 323(2)(1991), 877-895. https://doi.org/10.1090/S0002-9947-1991-1083144-2

T. Bartsch, Z. Liu, On a superlinear elliptic p-laplacian equation, J. Differ. Equ. 198(2004), 149-175. https://doi.org/10.1016/j.jde.2003.08.001

L. Brasco, E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9(2016), 323-355. https://doi.org/10.1515/acv-2015-0007

H. Brezis, P. G. Ciarlet,J. L. Lions, Analyse fonctionnelle: theorie et applications, Paris: Dunod, 1999. Print.

A. D. Castro, T. Kuusi, G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. Henri Poincare, Anal. Non Lineaire 33(5) (2016) 1279-1299. https://doi.org/10.1016/j.anihpc.2015.04.003

S. Cingolani, G. Vannella, Multiple positive solutions for a critical quasilinear equation via morse theory, Ann. I. H. Poincarrr'e 26(2009)397-413. https://doi.org/10.1016/j.anihpc.2007.09.003

G. Dinca, P. Jebelean, J. Mawhin, Variational and topological methods for dirichlet problems with p-laplacian Port. Math. (N. S.), 58 (2001), 339-378.

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47(2) (1974), 324-353. https://doi.org/10.1016/0022-247X(74)90025-0

J. F. Escobar, Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate, Commun. Pure Appl. Math. 43(7)(1990), 857-883. https://doi.org/10.1002/cpa.3160430703

G. Franzina, G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma (N. S.) 5 (2014) 373-386.

A. Ghanmi, K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Frac. Differ. Calculus, 6 (2016), 201-217. https://doi.org/10.7153/fdc-06-13

A. Ghanmi, Multiplicity of nontrivial solutions of a class of fractional p-Laplacian problem, Z. Anal. Anwend., 34 (2015), 309-319. https://doi.org/10.4171/ZAA/1541

M. Guedda,L. V'eron, Quasilinear elliptic equations involving critical sobolev exponents, Nonlinear Anal. Theory Methods Appl. 13(8) (1989), 879-902. https://doi.org/10.1016/0362-546X(89)90020-5

A. Iannizzotto, S. B. Liu, K. Perera, M. Squassina, Existence results for fractional p-Laplacian problem via morse theory, Adv. Calc. Var. 9(2) (2016), 101-125. https://doi.org/10.1515/acv-2014-0024

S. Jarohs, Strong comparison principle for the fractional p-Laplacian and applications to starshaped rings, Adv. Nonlinear Stud. 18(4) (2018) 691-704. https://doi.org/10.1515/ans-2017-6039

M. Kratou, Three solutions for a semilinear elliptic boundary value problem,Proc Math Sci 129, 22 (2019). https://doi.org/10.1007/s12044-019-0465-0

P. L. Lions, The concentration-compactness principle in the calculus of variations. the limit case, part 1. Rev. Mat. Iberoam. 1 (1985), 145-201. https://doi.org/10.4171/RMI/6

E. Lindgren, H¨older estimates for viscosity solutions of equations of fractional p-Laplace type, Nonlinear Differ. Equ. Appl. 23(5) (2016), 23-55. https://doi.org/10.1007/s00030-016-0406-x

X. Mingqi, B. Zhang, V.D. R'adulescu, Superlinear Schr¨odinger-Kirchhoff type problems involving the fractional pLaplacian and critical exponent, Adv. Nonlinear Anal. 9(1) (2020) 690-709. https://doi.org/10.1515/anona-2020-0021

G. Molica Bisci, V.D. R'adulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems, vol.162, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316282397

E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573. https://doi.org/10.1016/j.bulsci.2011.12.004

J. T. Schwartz, Generalizing the lusternik-schnirelman theory of critical points, Commun. Pure Appl. Math. 17(3)(1964), 307-315. https://doi.org/10.1002/cpa.3160170304

R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389(2) (2012), 887-898. https://doi.org/10.1016/j.jmaa.2011.12.032

M. Struwe, Three non-trival solutions of anticoercive boundary value problems for the pseudo-laplace-operator, Journal fr die reine und angewandte Mathematik, 325(1991), 68-74.

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differ. equ. 51(1 )(1984), 126-150. https://doi.org/10.1016/0022-0396(84)90105-0

M. Willem, Minimax theorems, PNLDE 24, Birkh¨auser, Boston-Basel-Berlin 1996. https://doi.org/10.1007/978-1-4612-4146-1

Z. Zhang, J. Chen, S. Li, Construction of pseudo-gradient vector field and sign- changing multiple solutions involving p-laplacian, J. Diffe. Equ. 201(2) (2004), 287-303. https://doi.org/10.1016/j.jde.2004.03.019

Publicado
2022-12-27
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Artigos