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  • Cited by 426
Publisher:
Cambridge University Press
Online publication date:
March 2016
Print publication year:
2016
Online ISBN:
9781316282397

Book description

This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.

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Contents

Bibliography
[1] N., Abatangelo and E., Valdinoci. A notion of nonlocal curvature. Numer. Funct. Anal. Optim. 35: 793–815, 2014.
[2] B., Abdellaoui, E., Colorado, and 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. Diff. Equations 11: 667–720, 2006.
[3] R. A., Adams. Sobolev Spaces. Academic Press, New York, 1975.
[4] S., Agmon. Lectures on elliptic boundary value problems. Mathematical Studies, Vol. 2. Van Nostrand, Princeton, NJ, 1965.
[5] S., Alama. Semilinear elliptic equations with sublinear indefinite nonlinearities. Adv. Diff. Equations 4: 813–42, 1999.
[6] F. J., Almgren and E. H., Lieb. Symmetric decreasing rearrangement is sometimes continuous. J. Am. Math. Soc. 2: 683–773, 1989.
[7] H., Amann and E., Zehnder. Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa 7: 539–603, 1980.
[8] A., Ambrosetti, H., Brézis, and G., Cerami. Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122: 519–543, 1994.
[9] A., Ambrosetti and A., Malchiodi. Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge Studies Advanced Mathematics 104. Cambridge University Press, 2007.
[10] A., Ambrosetti and G., Prodi, A Primer of Nonlinear Analysis. Cambridge Studies in Advanced Mathematics 34. Cambridge University Press, 1993.
[11] A., Ambrosetti and P. H., Rabinowitz. Dual variational methods in critical point theory and applications. J. Funct. Anal. 14: 349–81, 1973.
[12] A., Ambrosetti and M., Struwe. A note on the problem -Δu =λu+u|u|2*-2. Manusc. Math. 54: 373–79, 1986.
[13] D., Applebaum. Lévy Processes and Stochastic Calculus, 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge University Press, 2009.
[14] G., Arioli, F., Gazzola, H.-Ch., Grunau, and E., Sassone. The second bifurcation branch for radial solutions of the Brezis-Nirenberg problem in dimension four. NoDEA Nonlinear Differential Equations Appl. 15: 69–90, 2008.
[15] N., Aronszajn. Boundary values of functions with finite Dirichlet integral. Technical Report 14, University of Kansas, 1955, pp. 77–94.
[16] G., Autuori, A., Fiscella, and P., Pucci. Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity. Nonlinear Anal. (in press).
[17] G., Autuori and P., Pucci. Elliptic problems involving the fractional Laplacian in RN. J. Differential Equations 255: 2340–62, 2013.
[18] M., Badiale and E., Serra. Semilinear Elliptic Equations for Beginners. Springer, Berlin, 2011.
[19] A., Bahri and H., Berestycki. A perturbation method in critical point theory and applications. Trans. Am. Math. Soc. 267: 1–32, 1981.
[20] A., Bahri and P. L., Lions. Morse index of some min-max critical points. I. Application to multiplicity results. Commun. Pure Appl. Math. 41: 1027–37, 1988.
[21] G., Barles, E., Chasseigne, and C., Imbert. On the Dirichlet problem for second-order elliptic integrodifferential equations. Indiana Univ. Math. J. 57: 213–46, 2008.
[22] B., Barrios, E., Colorado, A. De, Pablo, and U., Sánchez. On some critical problems for the fractional Laplacian operator. J. Differential Equations 252: 6133–62, 2012.
[23] B., Barrios, E., Colorado, R., Servadei, and F., Soria. A critical fractional equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré Anal. Non Linéaire 31: 875–900, 2015.
[24] B. Barrios, Barrera, A., Figalli, and E., Valdinoci. Bootstrap regularity for integrodifferential operators and its application to nonlocal minimal surfaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, doi: 10.2422/2036-2145.201202_007.
[25] B., Barrios, M., Medina, and I., Peral. Some remarks on the solvability of non-local elliptic problems with the Hardy potential. Commun. Contemp. Math. 16(4): 1350046, 2014.
[26] P., Bartolo, V., Benci, and D., Fortunato. Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7: 981–1012, 1983.
[27] R., Bartolo, A. M., Candela, and A., Salvatore. Perturbed asymptotically linear problems. Ann. Mat. Pura Appl. 193: 89–101, 2014.
[28] R., Bartolo and G. Molica, Bisci. A pseudo-index approach to fractional equations. Expo. Math. 33: 502–516, 2015.
[29] T., Bartsch, Z., Liu, and T., Weth. Sign changing solutions of superlinear Schrödinger equations. Comm. Partial Differential Equations 29: 25–42, 2005.
[30] T., Bartsch, A., Pankov, and Z.-Q., Wang. Nonlinear Schrödinger equations with steep potential well. Commun. Contemp. Math. 4: 981–1012, 2001.
[31] T., Bartsch and Z.-Q., Wang. Existence and multiplicity results for some superlinear elliptic problems in RN . Commun. Partial Differential Equations 20: 1725–41, 1995.
[32] W., Beckner. Sobolev inequalities, the Poisson semigroup, and analysis on the sphere on SN . Proc. Natl. Acad. Sci. USA 89: 4816–19, 1992.
[33] V., Benci. On the critical point theory for indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 274: 533–72, 1982.
[34] S., Bernstein. Sur une classe d’équations fonctionnelles aux dérivées partielles. In Russian with French summary. Bull. Acad. Sci. URSS, Set. Math. 4: 17–26, 1940.
[35] J., Bertoin. Lévy Processes (Cambridge Tracts in Mathematics 121). Cambridge University Press, 1996.
[36] Z., Binlin, G. Molica, Bisci, and R., Servadei. Superlinear nonlocal fractional problems with infinitely many solutions. Nonlinearity 28: 2247–64, 2015.
[37] C., Bjorland, L., Caffarelli, and A., Figalli. Non-local gradient dependent operators. Adv. Math. 230: 1859–94, 2012.
[38] R. M., Blumenthal, R. K., Getoor, and D. B., Ray. On the distribution of first hits for the symmetric stable processes. Trans. Am. Math. Soc. 99: 540–54, 1961.
[39] L., Boccardo, M., Escobedo, and I., Peral. A Dirichlet problem involving critical exponents. Nonlinear Anal. 24: 1639–48, 1995.
[40] L., Boccardo and T., Gallouet. Problèmes unilatéraux avec données dans L1 (Unilateral problems with L1 data). C. R. Acad. Sci., Paris, Sér. I, 311: 617–19, 1990.
[41] M., Bonforte, Y., Sire, and J. L., Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Preprint available at http://arxiv. org/pdf/1404.6195v3.pdf.
[42] C., Brändle, E., Colorado, A. De, Pablo, and U., Sánchez. A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A, 143: 39–71, 2013.
[43] H., Brézis. Analyse fonctionelle: Théorie et applications. Masson, Paris, 1983.
[44] H., Brézis, J. M., Coron, and L., Nirenberg. Free vibrations for a nonlinear wave equation and a theorem of P. Rabinowitz. Comm. Pure Appl. Math. 33: 667–84, 1980.
[45] H., Brézis and E., Lieb. A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88: 486–90, 1983.
[46] H., Brézis and L., Nirenberg. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36: 437–77, 1983.
[47] C., Bucur and E., Valdinoci. Nonlocal diffusion and applications. Preprint available at http://arxiv.org/pdf/1504.08292v2.pdf.
[48] X., Cabré and Y., Sire. Nonlinear equations for fractional Laplacians: I. Regularity, maximum principles, and Hamiltonian estimates. Ann. Inst. H. Poincaré Anal. Non Linéaire 31: 23–53, 2014.
[49] X., Cabré and Y., Sire. Nonlinear equations for fractional Laplacians: II. Existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367: 911–41, 2015.
[50] X., Cabré and J., Solà-Morales. Layer solutions in a half-space for boundary reactions. Commun. Pure Appl. Math. 58: 1678–1732, 2005.
[51] X., Cabré and J., Tan. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224: 2052–93, 2010.
[52] L., Caffarelli. Nonlocal equations, drifts and games. In Nonlinear Partial Differential Equations, Abel Symposia 7: 37–52, 2012.
[53] L., Caffarelli, J. M., Roquejoffre, and Y., Sire. Variational problems with free boundaries for the fractional laplacian. J. Eur. Math. Soc. 12: 1151–79, 2010.
[54] L., 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: 425–61, 2008.
[55] L., Caffarelli and L., Silvestre. An extension problem related to the fractional Laplacian. Commun. Partial Differential Equations 32: 1245–60, 2007.
[56] L., Caffarelli and L., Silvestre. Regularity theory for fully nonlinear integrodifferential equations. Commun. Pure Appl. Math. 62: 597–638, 2009.
[57] L., Caffarelli and L., Silvestre. Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200: 59–88, 2011.
[58] A., Capella. Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains. Commun. Pure Appl. Anal. 10: 1645–62, 2011.
[59] A., Capella, J., Dávila, L., Dupaigne, and Y., Sire. Regularity of radial extremal solutions for some non-local semilinear equations. Commun. Partial Differential Equations 36: 1353–84, 2011.
[60] A., Capozzi, D., Fortunato, and G., Palmieri. An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 2: 463–70, 1985.
[61] A., Castro. Metodos variacionales y analisis functional no linear. X Colóquio Colombiano de Matemáticas 1980.
[62] G., Cerami, D., Fortunato, and M., Struwe. Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents. Ann. Inst. H. Poincaré Anal. Non Linéaire 1: 341–50, 1984.
[63] G., Cerami, S., Solimini, and M., Struwe. Some existence results for superlinear elliptic boundary value problems involving critical exponents. J. Funct. Anal. 69: 289–306, 1986.
[64] S., Challal, A., Lyaghfouri, and J. F., Rodrigues. On the A-obstacle problem and the Hausdorff measure of its free boundary. Ann. Mat. Pura Appl. 191: 113–65, 2012.
[65] M., Chang. Ground sate solutions of asymptotically linear fractional Schrödinger equations. J. Math. Phys. 54, 061504, 2013.
[66] F., Charro, E., Colorado, and I., Peral. Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side. J. Differential Equations 246: 4221–48, 2009.
[67] Z., Chen, N., Shioji, and W., Zou. Ground state and multiple solutions for a critical exponent problem. NoDEA Nonlinear Differential Equations Appl. 19: 253–77, 2012.
[68] M., Cheng. Bound state for the fractional Schrödinger equation with unbounded potential. J. Math. Phys. 53, 043507, 2012.
[69] Ph. G., Ciarlet. Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia, 2013.
[70] M., Clapp and T., WethMultiple solutions for the Brezis–Nirenberg problem. Adv. Differential Equations 10: 463–80, 2005.
[71] D. C., Clark. A variant of the Lusternik–Schnirelman theory. Indiana Univ. Math. J. 22: 65–74, 1972.
[72] F., Colasuonno and P., Pucci. Multiplicity of solutions for p(x)-polyharmonic elliptic Kirchhoff equations. Nonlinear Anal. 74: 5962–74, 2011.
[73] E., Colorado, A. De, Pablo, and U., Sánchez. Perturbations of a critical fractional equation. Pacific J. Math. 271: 65–85, 2014.
[74] M., Comte. Solutions of elliptic equations with critical Sobolev exponent in dimension three. Nonlinear Anal. 17: 445–55, 1991.
[75] R., Cont and P., Tankov. Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series. Boca Raton, FL, 2004.
[76] F. J. S. A., Corrêa and D. G., Costa. On a bi-nonlocal p(x)-Kirchhoff equation via Krasnoselskii's genus. Math. Meth. Appl. Sci. 38: 87–93, 2014.
[77] F. J. S. A., Corrêa and G. M., Figueiredo. On a p-Kirchhoff equation via Krasnoselskii's genus. Appl. Math. Lett. 22: 819–22, 2009.
[78] C., Cortazar, M., Elgueta, J., Rossi, and N., Wolanski. Asymptotic behavior for nonlocal diffusion equations. J. Math. Pure Appl. 86: 271–91, 2006.
[79] C., Cortazar, M., Elgueta, J., Rossi, and N., Wolanski. Boundary fluxes for nonlocal diffusion. J. Differential Equations 234: 360–90, 2007.
[80] C., Cortazar, M., Elgueta, J., Rossi, and N., Wolanski. How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Rat. Mech. Anal. 187: 137–56, 2008.
[81] V. Coti, Zelati and M., Nolasco. Existence of ground states for nonlinear, pseudorelativistic Schrödinger equations. Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend. Lincei, Mat. Appl. 22: 51–72, 2011.
[82] A., Cotsiolis and N., Tavoularis. Best constants for Sobolev inequalities for higher order fractional derivatives. J. Math. Anal. Appl. 295: 225–36, 2004.
[83] E. Di, Nezza, G., Palatucci, and E., Valdinoci. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136: 521–73, 2012.
[84] S., Dipierro, G., Palatucci, and E., Valdinoci. Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian. Le Matematiche 68: 201–16, 2013.
[85] S., Dipierro, X., Ros-Oton, and E., Valdinoci. Nonlocal problems with Neumann boundary conditions. WIAS Preprint No. 1986, (2014). Preprint available at http://arxiv.org/pdf/1407. 3313v3.pdf.
[86] S., Dipierro, O., Savin, and E., Valdinoci. All functions are locally s-harmonic up to a small error. Preprint available at http://arxiv.org/pdf/1404.3652v1.pdf.
[87] G., Devillanova and S., Solimini. Concentration estimates and multiple solutions to elliptic problems at critical growth. Adv. Differential Equations 7: 1257–80, 2002.
[88] G., Devillanova and S., Solimini. A multiplicity result for elliptic equations at critical growth in low dimension. Commun. Contemp. Math. 5: 171–7, 2003.
[89] H., Dong and D., Kim. On L p-estimates for a class of non-local elliptic equations. J. Funct. Anal. 262: 1166–99, 2012.
[90] O., Druet. Elliptic equations with critical Sobolev exponents in dimension 3. Ann. Inst. H. Poincaré Anal. Non Linéaire 19: 125–42, 2002.
[91] L. C., Evans. Partial differential equations. In Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, RI, 1998.
[92] E. B., Fabes, C. E., Kenig, and R. P., Serapioni. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations 7: 77–116, 1982.
[93] M. M., Fall. Semilinear elliptic equations for the fractional Laplacian with Hardy potential. Preprint available at http://arxiv.org/pdf/1109.5530v4.pdf.
[94] M. M., Fall and V., Felli. Unique continuation property and local asympotics of solutions to fractional elliptic equations. Comm. Partial Differential Equations 39: 354–97, 2014.
[95] P., Felmer, A., Quaas, and J., Tan. Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A 142: 1237–62, 2012.
[96] M., Ferrara and G. Molica, Bisci. Some applications of a Pucci–Serrin result. In Minimax Theory and Its Applications (in press) Helderman, Germany.
[97] M., Ferrara, G. Molica, Bisci, and B., Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems Series B, 19: 2493–99, 2014.
[98] G. M., Figueiredo, G. Molica, Bisci, and R., Servadei. On a fractional Kirchhoff-type equation via Krasnoselskii's genus. Asymptot. Anal. 94: 347–361, 2015.
[99] A., Fiscella. Saddle point solutions for non-local elliptic operators. Topol. Methods Nonlinear Anal. 44: 527–38, 2014.
[100] A., Fiscella, G. Molica, Bisci, and R., Servadei. Bifurcation and multiplicity results for critical nonlocal fractional Laplacian problems. Bull. Sci. Math. 140: 13–35, 2016.
[101] A., Fiscella and P., Pucci. On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems. Preprint.
[102] A., Fiscella, R., Servadei, and E., Valdinoci. A resonance problem for non-local elliptic operators. Z. Anal. Anwendungen 32: 411–31, 2013.
[103] A., Fiscella, R., Servadei, and E., Valdinoci. Density properties for fractional Sobolev spaces. Ann. Acad. Sci. Fenn. Math. 40: 235–53, 2015.
[104] A., Fiscella, R., Servadei, and E., Valdinoci. Asymptotically linear problems driven by fractional Laplacian operators. Math. Methods Appl. Sci. 38: 3551–3563, 2015.
[105] A., Fiscella and E., Valdinoci. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94: 156–70, 2014.
[106] G. B., Folland. Real Analysis: Modern Techniques and Their Applications. Wiley, NewYork, 1984.
[107] G., Franzina and G., Palatucci. Fractional p-eigenvalues. Riv. Math. Univ. Parma 5: 373–86, 2014.
[108] R. L., Frank and E., Lenzmann. Uniqueness and nondegeneracy of ground states for (−Δ)sQ + Q - Qa+1 = 0 in R. Ann. Math. (in press).
[109] M. F., Furtado, L. A., Maia, and E. A. B., Silva. On a double resonant problem in RN . Differential Integral Equations 15: 1335–44, 2002.
[110] E., Gagliardo. Proprietà di alcune classi di funzioni in più variabili. Ricerche Mat. 7: 102–37, 1958.
[111] J., García-Azorero and I., Peral. Multiplicity of solutions for elliptic problems with critical exponent or with non-symetric term. Trans. Am. Math. Soc. 323: 877–95, 1991.
[112] F., Gazzola and H.-Ch., Grunau. On the role of space dimension n =2+2 v2 in the semilinear Brezis–Nirenberg eigenvalue problem. Analysis (Munich) 20: 395–9, 2000.
[113] F., Gazzola and V., Rădulescu. A nonsmooth critical point theory approach to some nonlinear elliptic equations in RN . Differential Integral Equations 13: 47–60, 2002.
[114] F., Gazzola and B., Ruf. Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations. Adv. Differential Equations 2: 555–72, 1997.
[115] R. K., Getoor. First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101: 75–90, 1961.
[116] N., Ghoussoub, D., Preiss, A general mountain pass principle for locating and classifying critical points. Ann. Inst. H. Poincaré Anal. Non Linéaire 6: 321–30, 1989.
[117] D., Gilbarg and N. S., Trudinger. Elliptic Partial Differential Equations of Second Order, 2nd Ed. Springer-Verlag, Berlin, 1983.
[118] P., Grisvard. Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985.
[119] L., Hörmander. The Analysis of Linear Partial Differential Operators, Vols. I and II. Springer- Verlag, Berlin, 1983.
[120] O., Kavian. Introduction à la théorie des points critiques et applications aux problèmes elliptiques. In Mathématiques & Applications. Springer-Verlag, Paris, 1993.
[121] B., Kawohl. Rearrangements and Convexity of Level Sets in PDE. Springer-Verlag, Berlin, 1985.
[122] G. R., Kirchhoff. Vorlesungen über Mathematische Physik: Mechanik. Teubner, Leipzig, 1883.
[123] M. A., Krasnoselskii. Topological Methods in the Theory of Nonlinear Integral Equations. Macmillan, New York, 1964.
[124] A., Kristály. A double eigenvalue problem for Schrödinger equations involving sublinear nonlinearities at infinity. Electron. J. Differential Equations 42: 1–11, 2007.
[125] A., Kristály, V., Rădulescu, and Cs., Varga. Variational principles in mathematical physics, geometry, and economics: Qualitative analysis of nonlinear equations and unilateral problems. In Encyclopedia of Mathematics and Its Applications, Vol. 136. Cambridge University Press, 2010.
[126] A., Kristály and Cs., Varga. Multiple solutions for elliptic problems with singular and sublinear potentials. Proc. Am. Math. Soc. 135: 2121–6, 2007.
[127] T., Kuusi, G., Mingione, and Y., Sire. Nonlocal equations with measure data. Commun. Math. Phys. 337: 1317–68, 2015.
[128] T., Kuusi, G., Mingione, and Y., Sire. Nonlocal self-improving properties. Anal. PDE 8: 57–114, 2015.
[129] N. S., Landkof. Foundations of Modern Potential Theory, trans. from Russian (Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 180). Springer-Verlag, Berlin, 1973.
[130] N., Laskin. Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268: 298– 305, 2000.
[131] N., Laskin. Fractional Schrödinger equation. Phys. Rev. E 66, 056108, 2002.
[132] E. H., Lieb and M., Loss. Analysis. American Mathematical Society, Providence, 1997.
[133] E., Lindgren and P., Lindqvist. Fractional eigenvalues. Calc. Var. 49: 795–826, 2014.
[134] J. L., Lions. On some questions in boundary value problems of mathematical physics. In Proceedings of International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, 1977. Math. Stud. 30: 284–346, 1978.
[135] P.-L., Lions. The concentration-compactness principle in the calculus of variations: The limit case I. Rev. Mat. Iberoam. 1: 145–201, 1985.
[136] P.-L., Lions. The concentration-compactness principle in the calculus of variations: The limit case II. Rev. Mat. Iberoam. 1: 45–121, 1985.
[137] J. L., Lions and E., Magenes. Problemi ai limiti non omogenei (III). Ann. Scuola Norm. Sup. Pisa 15: 41–103, 1961.
[138] J. L., Lions and E., Magenes. Problémes aux limites non homogénes et applications. In Travaux et Recherches Mathématiques. Dunod, Paris, 1968.
[139] R. de la, Llave and E., Valdinoci. A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 26: 1309–44, 2009.
[140] A., Majda and E., Tabak. A two-dimensional model for quasigeostrophic flow: comparision with the two-dimensional Euler flow. Nonlinear phenomena in ocean dynamics. Phys. D 98(2–4): 515–22, 1996.
[141] W., McLean. Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, 2000.
[142] N., Meyers and J., Serrin. H = W. Proc. Nat. Acad. Sci. USA 51: 1055–6, 1964.
[143] L., Modica. A gradient bound and a Liouville theorem for nonlinear Poisson equations. Commun. Pure Appl. Math. 38: 679–84, 1985.
[144] G. Molica, Bisci. Fractional equations with bounded primitive. Appl. Math. Lett. 27: 53–8, 2014.
[145] G. Molica, Bisci. Sequence of weak solutions for fractional equations. Math. Res. Lett. 21: 241–53, 2014.
[146] G. Molica, Bisci and B. A., Pansera. Three weak solutions for nonlocal fractional equations. Adv. Nonlinear Stud. 14: 619–29, 2014.
[147] G. Molica, Bisci and V., Rădulescu. Multiplicity results for elliptic fractional equations with subcritical term. NoDEA Nonlinear Differential Equations Appl. 22: 721–739, 2015.
[148] G. Molica, Bisci and V., Rădulescu. A sharp eigenvalue theorem for fractional elliptic equations. Preprint 2015.
[149] G. Molica, Bisci and V., Rădulescu. Ground state solutions of scalar field fractional Schrödinger equations. Calc. Var. Partial Differential Equations 54: 2985–3008, 2015
[150] G. Molica, Bisci and D., Repovš. Existence and localization of solutions for nonlocal fractional equations. Asymptot. Anal. 90: 367–78, 2014.
[151] G. Molica, Bisci and D., Repovš. Fractional nonlocal problems involving nonlinearities with bounded primitive. J. Math. Anal. Appl. 420: 167–76, 2014.
[152] G. Molica, Bisci and D., Repovš. On doubly nonlocal fractional elliptic equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26: 161–76, 2015.
[153] G. Molica, Bisci, D., Repovš, and R., Servadei. Nontrivial solutions of superlinear nonlocal problems. Forum Math., to appear.
[154] G. Molica, Bisci, D., Repovš, and L., Vilasi. Integrodifferential fractional problems with infinitely many solutions. Preprint 2015.
[155] G. Molica, Bisci and R., Servadei. A bifurcation result for nonlocal fractional equations. Anal. Appl. 13: 371–94, 2015.
[156] G. Molica, Bisci and R., Servadei. Lower semicontinuity of functionals of fractional type and applications to nonlocal equations with critical Sobolev exponent. Adv. Differential Equations 20: 635–60, 2015.
[157] G. Molica, Bisci and R., Servadei. A Brezis–Nirenberg splitting approach for nonlocal fractional equations. Nonlinear Anal. 119: 341–53, 2015.
[158] G. Molica, Bisci and F., Tulone. An existence result for fractional Kirchhoff-type equations. Z. Anal. Anwendungen (in press).
[159] G. Molica, Bisci and L., Vilasi. On a fractional degenerate Kirchhoff-type problem. Commun. Contemp. Math. DOI: 10.1142/S0219199715500881.
[160] E., Montefusco, B., Pellacci, and G., Verzini. Fractional diffusion with Neumann boundary conditions: the logistic equation. Disc. Cont. Dyn. Syst. Ser. B 18: 2175–202, 2013.
[161] R., Musina and A. I., Nazarov. On fractional Laplacians. Commun. Partial Differential Equations 39: 1780–90, 2014.
[162] G., Palatucci and A., Pisante. Improved Sobolev embeddings, profile decomposition and concentration-compactness for fractional Sobolev spaces. Calc. Var. Partial Differential Equations 50: 799–829, 2014.
[163] Y. J., Park. Fractional Polya-Szegö inequality. J. Chungcheong Math. Soc. 24: 267–71, 2011.
[164] K., Perera, M., Squassina, and Y., Yang. Bifurcation and multiplicity results for critical fractional p-Laplacian problems. Math. Nachr. (in press).
[165] P., Piersanti and P., Pucci. Existence theorems for fractional p-Laplacian problems. Anal. Appl. (Singap.), in press. Preprint 2015.
[166] S., Pohozaev. On a class of quasilinear hyperbolic equations. Math. Sborniek 96:152–66, 1975.
[167] P., Pucci and S., Saldi. Critical stationary Kirchhoff equations in Rn involving nonlocal operators. Rev. Mat. Iberoam. (in press).
[168] P., Pucci and S., Saldi. Multiple solutions for an eigenvalue problem involving non-local elliptic p-Laplacian operators. In Geometric Methods in PDEs (Springer INdAM Series), Vol. 13, ed. by G., Citti, M., Manfredini, D., Morbidelli, S., Polidoro, and F., Uguzzoni. Springer-Verlag, Berlin, 2015, pp. 159–76.
[169] P., Pucci and J., Serrin. A mountain pass theorem. J. Differential Equations 60:142–9, 1985.
[170] P., Pucci, M., Xiang, and B., Zhang. Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal., in press. Preprint 2015.
[171] P., Pucci, M., Xiang, and B., Zhang. Multiple solutions for nonhomogeneous Schrödinger- Kirchhoff type equations involving the fractional p-Laplacian in RN . Calc. Var. Partial Differential Equations (in press).
[172] P. H., Rabinowitz. Some critical point theorems and applications to semilinear elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4: 215–23, 1978.
[173] P. H., Rabinowitz. Some minimax theorems and applications to nonlinear partial differential equations. In Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe, ed. by L., Cesari et al. New York: Academic Press, 1978, pp. 161–77.
[174] P. H., Rabinowitz. Minimax methods in critical point theory with applications to differential equations (CBMS Reg. Conf. Ser. Math. 65). American Mathematical Society, Providence, RI, 1986.
[175] P. H., Rabinowitz. On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43: 270–91, 1992.
[176] B., Ricceri. A general variational principle and some of its applications. Fixed point theory with applications in nonlinear analysis. J. Comput. Appl. Math. 113: 401–10, 2000.
[177] B., Ricceri. On a three critical points theorem. Archiv der Mathematik 75: 220–6, 2000.
[178] B., Ricceri. A three critical points theorem revisited. Nonlinear Anal. 70: 3084–9, 2009.
[179] B., Ricceri. A further three critical points theorem. Nonlinear Anal. 71: 4151–7, 2009.
[180] B., Ricceri. Nonlinear eigenvalue problems. In D. Y., Gao and D., Motreanu (eds.), Handbook of Nonconvex Analysis and Applications. International Press, 2010, pp. 543–95.
[181] B., Ricceri. A multiplicity result for nonlocal problems involving nonlinearities with bounded primitive. Stud. Univ. Babęs-Bolyai, Math. 55: 107–14, 2010.
[182] B., Ricceri. A further refinement of a three critical points theorem. Nonlinear Anal. 74: 7446– 54, 2011.
[183] B., Ricceri. A new existence and localization theorem for Dirichlet problem. Dynam. Systems Appl. 22: 317–24, 2013.
[184] X., Ros-Oton and J., Serra. The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pure Appl. 101: 275–302, 2014.
[185] X., Ros-Oton and J., Serra. The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213: 587–628, 2014.
[186] X., Ros-Oton and J., Serra. Fractional Laplacian: Pohozaev identity and nonexistence results. Preprint available at http://arxiv.org/pdf/1205.0494.pdf.
[187] S., Salsa. Partial Differential Equations in Action: From Modelling to Theory. Springer- Verlag Italia, Milano, 2008.
[188] S., Salsa. Optimal regularity in lower dimensional obstacle problems. In Subelliptic PDEs and Applications to Geometry and Finance (Lect. Notes Semin. Interdiscip. Mat. 6, Semin. Interdiscip. Mat. (S.I.M.)). Potenza, 2007, pp. 217–26.
[189] S., Salsa. The problems of the obstacle in lower dimension and for the fractional Laplacian. In Regularity Estimates for Nonlinear Elliptic and Parabolic Problems (Lecture Notes in Math. 2045). Springer, Heidelberg, 2012, pp. 153–244.
[190] M., Schechter. A variation of the mountain pass lemma and applications. J. Lond. Math. Soc. 44: 491–502, 1991.
[191] M., Schechter and W., Zou. On the Brezis–Nirenberg problem. Arch. Ration. Mech. Anal. 197: 337–56, 2010.
[192] S., Secchi. Ground state solutions for nonlinear fractional Schröinger equations in RN . J. Math. Phys. 54, 031501, 2013.
[193] S., Secchi. Perturbation results for some nonlinear equations involving fractional operators. Differ. Equ. Appl. 5: 221–36, 2013.
[194] S., Secchi. On fractional Schrödinger equations in RN without the Ambrosetti–Rabinowitz condition. Topol. Methods Nonlinear Anal. (in press).
[195] R., Servadei. Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity. Contemp. Math. 595: 317–40, 2013.
[196] R., Servadei. The Yamabe equation in a non-local setting. Adv. Nonlinear Anal. 2: 235–70, 2013.
[197] R., Servadei. A critical fractional Laplace equation in the resonant case. Topol. Methods Nonlinear Anal. 43: 251–67, 2014.
[198] R., Servadei and E., Valdinoci. Mountain pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389: 887–98, 2012.
[199] R., Servadei and E., Valdinoci. Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators. Rev. Mat. Iberoam. 29: 1091–126, 2013.
[200] R., Servadei and E., Valdinoci. Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33: 2105–37, 2013.
[201] R., Servadei and E., Valdinoci. A Brezis–Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Anal. 12(6): 2445–64, 2013.
[202] R., Servadei and E., Valdinoci. Weak and viscosity solutions of the fractional Laplace equation. Publ. Mat. 58: 133–54, 2014.
[203] R., Servadei and E., Valdinoci. On the spectrum of two different fractional operators. Proc. Roy. Soc. Edinburgh Sect. A. 144A: 1–25, 2014.
[204] R., Servadei and E., Valdinoci. The Brezis–Nirenberg result for the fractional Laplacian. Trans. Am. Math. Soc. 367: 67–102, 2015.
[205] R., Servadei and E., Valdinoci. Fractional Laplacian equations with critical Sobolev exponent. Rev. Mat. Comput. 28: 655–676, 2015.
[206] L., Silvestre. Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60: 67–112, 2007.
[207] L. N., Slobodeckij. Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations. Leningrad. Gos. Ped. Inst. Učep. Zap. 197: 54–112, 1958.
[208] P. R., Stinga and J. L., Torrea. Extension problem and Harnack's inequality for some fractional operators. Commun. Partial Differ. Equations 35: 2092–122, 2010.
[209] P., Stinga and B., Volzone. Fractional semilinear Neumann problems arising from a fractional Keller-Segel model. Calc. Var. Partial Differential Equations 54: 1009–42, 2015.
[210] W. A., Strauss. Existence of solitary waves in higher dimensions. Commun. Math. Phys. 55: 149–62, 1977.
[211] M., Struwe. Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems. Manuscripta Math. 32: 335–64, 1980.
[212] M., Struwe. Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (Ergebnisse der Mathematik und ihrer Grenzgebiete 3). Springer- Verlag, Berlin, 1990.
[213] A., Szulkin, T., Weth, and M., Willem. Ground state solutions for a semilinear problem with critical exponent. Differential Integral Equations 22: 913–26, 2009.
[214] J., Tan. The Brezis–Nirenberg type problem involving the square root of the Laplacian. Calc. Var. Partial Differential Equations 36: 21–41, 2011.
[215] K., Teng. Multiple solutions for a class of fractional Schrödinger equations in RN. Nonlinear Anal. Real World Appl. 21: 76–86, 2015.
[216] E., Valdinoci. From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. S'eMA 49: 33–44, 2009.
[217] J. L., Vázquez. Nonlinear diffusion with fractional laplacian operators. In Nonlinear Partial Differential Equations, Abel Symp. 7: 271–98, 2012.
[218] J. L., Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete Contin. Dyn. Syst. Ser. S 7(4): 857–85, 2014.
[219] L., Vlahos, H., Isliker, Y., Kominis, and K., Hizonidis. Normal and anomalous diffusion: a tutorial In Order and Chaos, Vol. 10, ed. by T., Bountis. Patras University Press, 2008.
[220] M., Willem. Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications 24). Birkhäuser, Boston, 1996.
[221] D., Zhang. On multiple solutions of Δu + λu +|u|4/(n−2)u = 0. Nonlinear Anal. 13: 353–72, 1989.

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