Existence, multiplicity, perturbation, and concentration results for a class of quasi-linear elliptic problems
DOI:
https://doi.org/10.58997/ejde.mon.07Keywords:
Critical points; non-smooth functionals; weak slope; deformation theorem; invariant functionals, mountain pass theorem; relative categoryAbstract
The aim of this monograph is to present a comprehensive survey of results about existence, multiplicity, perturbation from symmetry and concentration phenomena for a class of quasi-linear elliptic equations coming from functionals of the calculus of variations which turn out to be merely continuous. Some tools of non-smooth critical point theory will be employed.
For more information see https://ejde.math.txstate.edu/Monographs/07/abstr.html
References
Adimurthi, Critical points for multiple integrals of the calculus of variations, Ann. Sc. Norm. Sup. Pisa 17 (1990), 393–413.
A. Ambrosetti, A perturbation theorem for superlinear boundary value problems, Math. Res. Center, Univ. Wisconsin-Madison, Tech. Sum. Report 1446 (1974).
A. Ambrosetti, M. Badiale, S. Cingolani, Semiclassical states of nonlinear Schr¨odinger equations, Arch. Ration. Mech. Anal. 140 (1997), 285–300.
A. Ambrosetti, G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl. 93 (1972), 231–247.
A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
D. Arcoya, L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rat. Mech. Anal. 134 (1996), 249–274.
, Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl. 6 (1999), 79–100.
G. Arioli, F. Gazzola, Weak solutions of quasilinear elliptic PDE’s at resonance, Ann. Fac. Sci. Toulouse 6 (1997), 573–589.
, Existence and multiplicity results for quasilinear elliptic differential systems, Comm. Partial Differential Equations 25 (2000), 125–153
, Quasilinear elliptic equations at critical growth, NoDEA, Nonlinear Differential Equations Appl. 5 (1998), 83–97.
, Some results on p-Laplace equations with a critical growth term, Differential Integral Equations 11 (1998), 311–326.
F.V. Atkinson, L.A. Peletier, Elliptic equations with nearly critical rowth, J. Differential Equations 70 (1987), 349–365.
J.P. Aubin, I. Ekeland, Applied Nonlinear Analysis, Wiley-Interscience, New York, 1984.
M. Badiale, G. Citti, Concentration compactness principle and quasilinear elliptic equations in Rn, Comm. Partial Differential Equations 16 (1991), 1795–1818.
A. Bahri, H. Berestyki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1981), 1–32.
A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253–294.
A. Bahri, P.L. Lions, Morse index of some min-max critical points. I. Applications to multiplicity results, Comm. Pure Appl. Math. 41 (1988), 1027–1037.
A. Bahri, P.L. Lions, Solutions of superlinear elliptic equations and their Morse indices, Comm. Pure Appl. Math. 45 (1992), 1205–1215.
V. Benci, G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal. 99 (1987), 283–300.
H. Berestycki, P.L. Lions, Nonlinear scalar field equation I, existence of a ground state, Arch. Rat. Mech. Anal. 82 (1983), 313–346.
L. Boccardo, The Bensoussan & Co. technique for the study of some critical points problems, Optimal Control and PDE, J.L. Menaldi et al. ed. Book in honour of Professor Alain Bensoussan’s 60th birthday, IOS Press (2000).
L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal. 19 (1992), 581–597.
L. Boccardo, B. Pellacci, Bounded positive critical points of some multiple integrals of the calculus of variations, Progr. Nonlinear Differential Equations Appl. 54, Birkh¨auser, Basel, 2003, 33–51.
P. Bolle, On the Bolza Problem, J. Differential Equations 152 (1999), 274–288.
P. Bolle, N. Ghoussoub, H. Tehrani, The multiplicity of solutions in nonhomogeneous boundary value problems, Manuscripta Math. 101 (2000), 325–350.
H. Brezis, Probl`emes unilat´eraux, J. Math. Pures Appl. 51 (1972), 1–68.
H. Br´ezis, F.E. Browder, Sur une propri´et´e des espaces de Sobolev, C. R. Acad. Sc. Paris 287 (1978), 113–115.
H. Br´ezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437–477.
A.M. Candela, A. Salvatore, Multiplicity results of an elliptic equation with non homogeneous boundary conditions, Topol. Meth. Nonlinear Anal. 11 (1998), 1–18.
A.M. Candela, A. Salvatore, M. Squassina, Multiple solutions for semilinear elliptic systems with nonhomogeneous boundary conditions, Nonlinear Anal. 51 (2002), 249–270.
A. Canino, On a jumping problem for quasilinear elliptic equations, Math. Z. 226 (1997), 193–210.
A. Canino, On a variational approach to some quasilinear problems, Serdica Math. J. 22 (1996), 399–426.
A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Meth. Nonlinear Anal. 6 (1995), 357–370.
A. Canino, On the existence of three solutions for jumping problems involving quasilinear operators, Topol. Meth. Nonlinear Anal. 18 (2001), 1–16.
A. Canino, Variational bifurcation for quasilinear elliptic equations, Calc. Var. Partial Differential Equations 18 (2003), 269–286.
A. Canino, M. Degiovanni, Nonsmooth critical point theory and quasilinear elliptic equations Topological Methods in Differential Equations and Inclusions, 1–50 - A. Granas, M. Frigon, G. Sabidussi Eds. - Montreal (1994), NATO ASI Series - Kluwer A.P. (1995).
D. Cao, X. Zhu, The concentration–compactness principle in nonlinear elliptic equations, Acta Math. Sci. 9 (1989), 307–328.
G. Cerami, D. Fortunato, M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. Henri Poincare Anal. Nonlin. 1 (1984), 341–350.
J. Chabrowski, Variational methods for potential operator equations, Walter de Gruyter Studies in Mathematics 24, Walter de Gruyter & Co., Berlin - New York, 1997.
J. Chabrowski, Weak Convergence Methods for Semilinear Elliptic Equations, World Scientific, Singapore, New Jersey, London, Hong Kong, 1999.
J. Chabrowski, On multiple solutions for the nonhomogeneous p-Laplacian with a critical Sobolev exponent, Differential Integral Equations, 8 (1995), 705–716.
K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, 6. Birkh¨auser, (1993).
, Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.
C.C. Chen, C.S. Lin, Uniqueness of the ground state solutions of u + f(u) = 0 in RN, N 3, Comm. Partial Differential Equations 16 (1991), 1549–1572.
G. Citti, Existence of positive solutions of quasilinear degenerate elliptic equations on unbounded domains, Ann. Mat. Pura Appl. (4) 158 (1991), 315–330.
M. Clapp, Critical point theory for perturbations of symmetric functionals, Comment. Math. Helv. 71 (1996), 570–593.
F.H. Clarke, I. Ekeland, Nonlinear oscillations and boundary value problems for Hamiltonian systems, Arch. Rat. Mech. Anal. 78 (1982), 315–333.
M. Conti, F. Gazzola, Positive entire solutions of quasilinear elliptic problems via nonsmooth critical point theory, Topol. Meth. Nonlinear Anal. 8 (1996), 275–294.
J.N. Corvellec, M. Degiovanni, Nontrivial solutions of quasilinear equations via nonsmooth Morse theory, J. Differential Equations 136 (1997), 268–293.
J.N. Corvellec, M. Degiovanni, M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Meth. Nonlinear Anal. 1 (1993), 151–171.
V. Coti Zelati, P. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1991), 693–727.
R. Courant, D. Hilbert, Methods of Mathematical Physics, Vols. I and II, Interscience, New York, (1953) and (1962).
B. Dacorogna, Direct methods in the calculus of variations, Springer Verlag, (1988).
G. Dal Maso, F. Murat, Almost everywhere convergence of gradients of solutions to nonlinear elliptic systems, Nonlinear Anal. 31 (1998), 405–412.
L. Damascelli, M. Ramaswamy, Symmetry of C1 solutions of p-Laplace equations in RN, Adv. Nonlinear Stud. 1 (2001), 40–64.
E.N. Dancer, A note on an equation with critical exponent, Bull. London Math. Soc. 20 (1988), 600–602.
E. De Giorgi, Un esempio di estremali discontinui per un problema variazionale di tipo ellittico, Boll. UMI 4 (1968), 135–137.
M. Degiovanni, M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl. (6) 167 (1994), 73–100.
M. Degiovanni, A. Musesti, M. Squassina, On the regularity of solutions in the Pucci-Serrin identity, Calc. Var. Partial Differential Equations 18 (2003), 317–334.
M. Degiovanni, M. Marzocchi, V. R˘adulescu, Multiple solutions of hemivariational inequalities with area-type term, Calc. Var. Partial Differential Equations 10 (2000), 355–387.
M. Degiovanni, S. Zani, Euler equations involving nonlinearities without growth conditions, Potential Anal. 5 (1996), 505–512.
M. Del Pino, P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.
, Multi-peak bound states for nonlinear Schr¨odinger equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 15 (1998), 127–149.
, Semi-classical states for nonlinear Schr¨odinger equations, J. Functional Anal. 149 (1997), 245–265.
E. DiBenedetto, C1, local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal. 7 (1983), 827–850.
G-C. Dong, S. Li, On the existence of infinitely many solutions of the dirichlet problem for some nonlinear elliptic equation, Univ. of Wisconsin Math. Research Center Teach. Rep. N02161, Dec. (1980).
I. Ekeland, N. Ghoussoub, H. Tehrani, Multiple solutions for a classical problem in the calculus of variations, J. Differential Equations 131 (1996), 229–243.
A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schr¨odinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.
B. Franchi, E. Lanconelli, J. Serrin, Existence and uniqueness of ground state solutions of quasilinear elliptic equations, Springer-Verlag, New York 1 (1988), 293–300.
J. Garcia Azorero, I. Peral Alonso, Existence and nonuniqueness for the p-Laplacian, Comm. Partial Differential Equations 12 (1987), 1389–1430.
, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 887–895.
, On limits of solutions of elliptic problems with nearly critical growth, Comm. Partial Differential Equations 17 (1992), 2113–2126.
N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge Tracts in Math. 107, Cambridge University Press, Cambridge 1993.
A. Giacomini, M. Squassina, Multi-peak solutions for a class of degenerate elliptic equations, Asymptotic Anal. 36 (2003), 115–147.
M. Giaquinta, Multiple integrals in the calculus of variations, Princeton University Press, Princeton, (1983).
B. Gidas, W.M. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
A. Groli, Jumping problems for quasilinear elliptic variational inequalities, NoDEA, Nonlinear Differential Equations Appl. 9 (2002), 117–147.
A. Groli, A. Marino, C. Saccon, Variational theorems of mixed type and asymptotically linear variational inequalities, Topol. Meth. Nonlinear Anal. 12 (1998), 109–136.
A. Groli, M. Squassina, On the existence of two solutions for a general class of jumping problems, Topol. Meth. Nonlin. Anal. 21 (2003), 325–344.
A. Groli, M. Squassina, Jumping problems for fully nonlinear elliptic variational inequalities, J. Convex Anal. 8 (2001), 471–488.
M. Guedda, L. Veron, Quasilinear elliptic problems involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879–902.
L. Jeanjean, K. Tanaka, A remark on least energy solutions in RN, Proc. Amer. Math. Soc. 131 (2003), 2399–2408.
, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, preprint, (2002).
H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493–514.
A. Ioffe, On lower semicontinuity of integral functionals I, Siam J. Control Optim. 15 (1977), 521–538.
A. Ioffe, E. Schwartzman, Metric critical point theory 1. Morse regularity and homotopic stability of a minimum, J. Math. Pures Appl. 75 (1996), 125–153.
G. Katriel, Mountain pass theorems and global homeomorphism theorems, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 11 (1994), 189–209.
J.L. Kazdan, J. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567–597.
M.C. Knaap, L.A. Peletier, Quasilinear elliptic equations with nearly critical growth, Comm. Partial Differential Equations 14 (1989), 1351–1383.
O.A. Ladyzhenskaya, N.N. Ural’tseva, Linear and Quasilinear Elliptic equations Nauka Press, Academic Press, New York, (1968).
S. Lancelotti, Morse index estimates for continuous functionals associated with quasilinear elliptic equations, Adv. Differential Equations 7 (2002), 99–128.
S. Lancelotti, A. Musesti, M. Squassina, Infinitely many solutions for poly-harmonic problems with broken symmetries, Math. Nachr. 253 (2003), 35–44.
G. Li, S. Yan, Eigenvalue problems for quasilinear elliptic equations in Rn, Comm. Partial Differential Equations. 14 (1989), 1291–1314.
P. Lindqvist, On the equation div(|ru|p−2ru) + |u|p−2u = 0, Proc. Amer. Math. Soc. 109 (1990), 157–162.
P.L. Lions, The concentration–compactness principle in the calculus of variations. The limit case, part I, Rev. Matem´atica Iberoamericana 1 (1985), 145–201.
P.L. Lions, The concentration–compactness principle in the calculus of variations. The limit case, part II, Rev. Matem´atica Iberoamericana 1 (1985), 45–121.
P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case, Ann. Inst. H. Poincar´e Anal. Non Lineaire 1 (1984), Part I, 109–145.
A. Marino, A. Micheletti, A. Pistoia, Some variationals results on semilinear elliptic problems with asymptotically nonsymmetric behaviour, Quaderni Sc. Norm. Sup. Pisa (1991), 243–256.
A. Marino, D. Passaseo, A jumping behavior induced by an obstacle, In M. Gilardi, M. Matzeu and F. Pacella, editors, Progress in Variational Methods in Hamiltonian Systems and Elliptic Equations, Pitman (1992), 127–143.
A. Marino, C. Saccon, Some variational theorems of mixed type and elliptic problems with jumping nonlinearities, Ann. Sc. Norm. Sup. di Pisa, 25 (1997), 631–665.
A. Marino, D. Scolozzi, Geodesics with obstacles, Boll. Un. Mat. Ital. B 2 (1983), 1–31.
M. Marzocchi, Multiple solutions of quasilinear equations involving an area-type term, J. Math. Anal. Appl. 196 (1995), 1093–1104.
J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems. Applied Mathematical
Sciences, 74, Springer-Verlag, New York, (1989).
C. Miranda, Alcune osservazioni sulla maggiorazione in L (Omega) delle soluzioni deboli delle equazioni ellittiche del secondo ordine, Ann. Mat. Pura Appl. 61 (1963), 151–168.
C.B. Morrey, Multiple integrals in the calculus of variations, Springer, (1966).
J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Math. J. 20 (1971), 1077–1092.
A. Musesti, M. Squassina, Asymptotics of solutions for fully nonlinear problems with nearly critical growth, Z. Anal. Anwendungen 21 (2002), 185–201.
Y.-G. Oh, Existence of semiclassical bound states of nonlinear Schrodinger equations with potentials of the class (V )a, Comm. Partial Differential Equations 13 (1988), 1499–1519.
Y.-G. Oh, Stability of semiclassical bound states of nonlinear Schr¨odinger equations with potentials, Comm. Math. Phys. 121 (1989), 11–33.
S. Paleari, M. Squassina, Multiplicity results for perturbed symmetric quasilinear elliptic systems, Differential Integral Equation 14 (2001), 785–800.
D. Passaseo, The effect of the domain shape on the existence of positive solutions of the equation u + u2 −1 = 0, Topol. Meth. Nonlinear Anal. 3 (1994), 27–54.
B. Pellacci, Critical points for non differentiable functionals, Boll. Un. Mat. Ital. B 11 (1997), 733–749.
B. Pellacci, Critical points for some functionals of the calculus of variations, Topol. Meth. Nonlinear Anal. 17 (2001), 285–305.
B. Pellacci, M. Squassina, Existence of unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations 201 (2004), 25–62.
S.I. Pohozaev, Eigenfunctions of the equations u + f(u) = 0, Soviet Math. Dokl. 6 (1965), 1408–1411.
P. Pucci, J. Serrin, A General Variational Identity, Indiana Math. J. 35 (1986), 681–703.
P. Rabier, C. Stuart, Exponential decay of the solutions of quasilinear second-order equations and Pohozaev identities, J. Differential Equations 165 (2000), 199–234.
P.H. Rabinowitz, Critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc. 272, (1982).
P.H. Rabinowitz, On a class of nonlinear Schrodinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Series Math. 65, Amer. Math. Soc., Providence, R.I. (1986).
P.H. Rabinowitz, Variational methods for nonlinear eigenvalue problems. Eigenvalues of non-linear problems (Centro Internaz. Mat. Estivo C.I.M.E.), III Ciclo, Varenna, 1974), pp. 139–195. Edizioni Cremonese, Rome, 1974.
J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.
J. Serrin, M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Univ. Math. J. 49 (2000), 897–923.
S. Solimini, Some remarks on the number of solutions of some nonlinear elliptic problems, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 2 (1985), 143–156.
G. Spradlin, A singularly perturbed elliptic partial differential equation with an almost periodic term, Calc. Var. Partial Differential Equations 9 (1999), 207–232.
G. Spradlin, An elliptic equation with spike solutions concentrating at local minima of the Laplacian of the potential, Electron. J. Differential Equations (2000), No. 32, 14 pp.
M. Squassina, Two solutions for inhomogeneous fully nonlinear elliptic equations at critical growth, NoDEA, Nonlinear Differential Equations Appl. 11 (2004), 53–71.
M. Squassina, Multiple critical points for perturbed symmetric functionals associated with quasilinear elliptic problems, Nonlinear Anal. 47 (2001), 1605–1616.
M. Squassina, Perturbed S1-symmetric Hamiltonian systems, Applied Math. Letters 14 (2001), 375–380.
M. Squassina, Existence of positive entire solutions for nonlinear elliptic problems, Topol. Meth. Nonlinear Anal. 17 (2001), 23–39.
M. Squassina, Spike solutions for a class of singularly perturbed quasilinear elliptic equations, Nonlinear Anal. 54 (2003), 1307-1336.
M. Squassina, Existence of weak solutions to general Euler’s equations via nonsmooth critical point theory, Ann. Fac. Sci. Toulouse (6) 9 (2000), 113–131.
M. Squassina, Existence of multiple solutions for quasilinear diagonal elliptic systems, Electron. J. Differential Equations 14 (1999), 1–12.
M. Squassina, On a class of nonlinear elliptic BVP at critical growth, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), 193–208.
G. Stampacchia, `Equations elliptiques du second ordre `a coefficients discontinus, S´eminaire de Math´ematiques Sup´erieures, Les Presses de l’Universit´e de Montr´eal, Montreal, Que. 16
(1966), 326pp.
M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer, Berlin (1996).
M. Struwe, Quasilinear elliptic eigenvalue problems, Comment. Math. Helvetici 58, (1983), 509–527.
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353–372.
K. Tanaka, Morse indices at critical points related to the symmetric mountain pass theorem and applications, Comm. Partial Differential Equations 14 (1989), 99–128.
G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 9 (1992), 281–304.
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), 126–150.
N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747.
L.S. Yu, Nonlinear p-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc. 115 (1992), 1037–1045.
A. Visintin, Strong convergence results related to strict convexity, Comm. Partial Differential Equations 9 (1984), 439–466.
X. Wang, On concentration of positive bound states of nonlinear Schr¨odinger equations, Comm. Math. Phys. 153 (1993), 229–244.
M. Willem, Minimax theorems, Birkhauser Verlag, Basel, 1996.
E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer-Verlag, New York, 1989.
H.S. Zhou, Solutions for a quasilinear elliptic equation with critical sobolev exponent and
perturbations on Rn, Differential Integral Equations 13 (2000), 595–612.
Downloads
Published
License
Copyright (c) 2023 Marco Squassina
This work is licensed under a Creative Commons Attribution 4.0 International License.
