Multiplicity of solutions for $p$-Laplacian type elliptic problems with electromagnetic fields and critical nonlinearity
Keywords
p-Laplacian type problem, magnetic fields, critical growth, concentration-compactness principle, variational methodAbstract
We consider a class of $p$-Laplacian type elliptic problems with electromagnetic fields and critical nonlinearity in bounded domains. New results about the existence and multiplicity of solutions to these problems are obtained by using the concentration-compactness principle and variational method.References
G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal. 170 (2003), 277–295.
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical exponents, Commun. Pure Appl. Math. 34 (1983), 437–477.
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
J. Chabrowski and A. Szulkin, On the Schrödinger equation involving a critical Sobolev exponent and magnetic field, Topol. Methods Nonlinear Anal. 25 (2005), 3–21.
J. Chabrowski, On multiple solutions for the nonhomogeneous p-Laplacian with a critical Sobolev exponent, Differential Integral Equations 8 (1995), 705–716.
J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. 3 (1995), 493–512.
J. Chen and S. Li, On multiple solutions of a singular quasi-linear equation on unbounded domain, J. Math. Anal. Appl. 275 (2002), 733–746.
S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field, J. Differential Equations 188 (2003), 52–79.
S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields, J. Math. Anal. Appl. 275 (2002), 108–130.
F.J.S.A. Corrêa and G.M. Figueiredo, On a elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74 (2006), 263–277.
F.J.S.A. Corrêa and R.G. Nascimento, On a nonlocal elliptic system of p-Kirchhofftype under Neumann boundary condition, Math. Comput. Modelling 49 (2009), 598–604.
G.W. Dai and R.F. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl. 359 (2009), 275–284.
M. Esteban and P.L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio De Giorgi, 1989, 369–408.
A. Ferrero and F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001), 494–522.
J. Garcia Azorero and I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), 441–476.
J. Garcia Azorero and I. Peral, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc. 323 (1991), 877–895.
N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), 5703–5743.
A. Hamydy, M. Massar and N. Tsouli, Existence of solutions for p-Kirchhoff type problems with critical exponent, Electron. J. Differential Equations, Vol. 2011 (2011), No. 105, pp. 1–8.
P. Han, Solutions for singular critical growth Schrödinger equation with magnetic field, Port. Math. 63 (2006), 37–45.
X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), 1407–1414.
X. He and W. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. 26 (2010), 387–394.
X. He and W. Zou, Infinitely many arbitrarily small solutions for sigular elliptic problems with critical Sobolev–Hardy exponents, Proc. Edinburgh Math. Soc. 52 (2009), 97–108.
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
M.A. Krasnosel’skiı̆, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Elmsford, NY, 1964.
K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal. 41 (2000), 763–778.
J.L. Lions, On some equations in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos., Inst. Mat. Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977; NorthHolland Math. Stud., vol. 30, North-Holland, Amsterdam, 1978, pp. 284–346.
J.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Parta I and II, Ann. Inst. H. Poincaré Anal. Non. Linéaire 1 (1984), 109–145, 223–283.
S. Li and W. Zou, Remarks on a class of elliptic problems with critical exponents, Nonlinear Anal. 32 (1998), 769–774.
D.C. Liu, On a p-Kirchhoff equation via Fountain theorem and dual fountain theorem, Nonlinear Anal. 72 (2010), 302–308.
T.F. Ma and J.E. Munoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), 243–248.
R. Kajikiya, A critical-point theorem related to the symmetric mountain-pass lemma and its applications to elliptic equations, J. Funct. Anal. 225 (2005), 352–370.
K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), 246–255.
P. H. Rabinowitz, Minimax methods in critical-point theory with applications to differential equations, CBME Regional Conference Series in Mathematics, Volume 65 (American Mathematical Society, Providence, RI, 1986).
E.A. Silva and M.S. Xavier, Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents, Annales Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 341–358.
Z. Tang, On the least energy solutions of nonlinear Schrödinger equations with electromagnetic fields, Comput. Math. Appl. 54 (2007), 627–637.
Z. Tang, Multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields and critical frequency, J. Differential Equations 245 (2008), 2723–2748.
F. Wang, On an electromagnetic Schrödinger equation with critical growth, Nonlinear Anal. 69 (2008), 4088–4098.
M. Willem, Minimax Theorems, Birkhäuser Boston, Boston, MA, 1996.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0