Abstract
We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator and a concave-convex term, together with mixed Dirichlet-Neumann boundary conditions.
Similar content being viewed by others
References
B. Abdellaoui, E. Colorado, I. Peral, Some remarks on elliptic equations with singular potentials and mixed boundary conditions. Adv. Nonlinear Stud. 4, No 4 (2004), 503–533.
B. Abdellaoui, E. Colorado, I. Peral, Effect of the boundary conditions in the behavior of the optimal constant of some Caffarelli-Kohn-Nirenberg inequalities. Application to some doubly critical nonlinear elliptic problems. Adv. Differential Equations 11, No 6 (2006), 667–672.
A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, No 2 (1994), 519–543.
A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349–381.
S. Alama, Semilinear elliptic equations with sublinear indefinite nonlinearities. Adv. Differential Equations 4 (1999), 813–842.
B. Barrios, E. Colorado, A. De Pablo, U. Sánchez, On some critical problems for the fractional Laplacian operator. J. Differential Equations 252, No 11 (2012), 6133–6162.
B. Barrios, E. Colorado, R. Servadei, F. Soria, A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincarée Anal. Non Linéeaire 32, No 4 (2015), 875–900.
L. Boccardo, M. Escobedo, I. Peral, A Dirichlet problem involving critical exponents. Nonlinear Anal. 24, No 11 (1995), 1639–1648.
C. Brändle, E. Colorado, A. de Pablo, U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71.
X. Cabré, J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, No 5 (2010), 2052–2093.
L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32, No 7-9, (2007), 1245–1260.
J. Carmona, E. Colorado, T. Leonori, A. Ortega, Regularity of solutions to a fractional elliptic problem with mixed Dirichlet-Neumann boundary data. Adv. in Calculus of Variations 2020 (2020),; DOI: 10.1515/acv-2019-0029.
F. Charro, E. Colorado, I. Peral, Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side. J. Differential Equations 246 (2009), 4221–4248.
W. Chen, C. Li, B. Ou, Qualitative Properties of Solutions for an Integral Equation. Disc. & Cont. Dynamics Sys. 12 (2005), 347–354.
E. Colorado, A. Ortega, The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions. J. Math. Anal. Appl. 473, No 2 (2019), 1002–1025.
E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions. J. Funct. Anal. 199, No 2 (2003), 468–507.
L. Damascelli, F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains. Rev. Mat. Iberoamericana 20, No 1 (2004), 67–86.
J. Denzler, Bounds for the heat diffusion through windows of given area. J. Math. Anal. Appl. 217, No 2 (1998), 405–422.
E.B. Fabes, C.E. Kenig, R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7, No 1 (1982), 77–116.
J. García Azorero, I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term. Trans. Amer. Math. Soc. 323 (1991), 877–895.
J. García Azorero, J. Manfredi, I. Peral, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2, No 3 (2000), 385–404.
B. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, No 3 (1979), 209–243.
B. Gidas, J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6, No 8 (1981), 883–901.
P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1, No 2 (1985), 45–121.
J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I. Springer-Verlag, New York-Heidelberg(1972).
P.H. Rabinowitz, Some global results for nonlinear eigenvalue problems. J. Funct. Analysis 7 (1971), 487–513.
J. Serrin, A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304–318.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Carmona, J., Colorado, E., Leonori, T. et al. Semilinear Fractional Elliptic Problems with Mixed Dirichlet-Neumann Boundary Conditions. Fract Calc Appl Anal 23, 1208–1239 (2020). https://doi.org/10.1515/fca-2020-0061
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1515/fca-2020-0061