Bifurcation and multiplicity results for critical p-Laplacian problems
DOI:
https://doi.org/10.12775/TMNA.2015.093Keywords
p-Laplacian, critical nonlinearity, bifurcation, multiplicity, existence, abstract critical point theory, Z^2-cohomological index, pseudo-indexAbstract
We prove a bifurcation and multiplicity result that is independent of the dimension N for a critical p-Laplacian problem that is the analog of the Brezis-Nirenberg problem for the quasilinear case. This extends a result in the literature for the semilinear case p = 2 to all p in (1;infty). In particular, it gives a new existence result when N \le p^2. When p \neq 2 the nonlinear operator -\Delta_p has no linear eigenspaces, so our extension is nontrivial and requires a new abstract critical point theorem that is not based on linear subspaces. We prove a new abstract result based on a pseudoindex related to the Z^2-cohomological index that is applicable here.References
G. Arioli and F. Gazzola, Some results on p-Laplace equations with a critical growth term, Differential Integral Equations 11 (1998), 311-326.
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 (1983), 981-1012.
V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), 533-572.
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437-477.
D. Cao, S. Peng and S. Yan, Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth, J. Funct. Anal. 262 (2012), 2861-2902.
A. Capozzi, D. Fortunato and G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincare Anal. Non Lineaire 2 (1985), 463-470.
G. Cerami, D. Fortunato and M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 341-350.
M. Degiovanni and S. Lancelotti, Linking solutions for p-Laplace equations with nonlinearity at critical growth, J. Funct. Anal. 256 (2009), 3643-3659.
G. Devillanova and S. Solimini, Concentration estimates and multiple solutions to elliptic problems at critical growth, Adv. Differential Equations 7 (2002), 1257-1280.
H. Egnell, Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents, Arch. Rational Mech. Anal. 104 (1988), 57-77.
E. R. Fadell and P. H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978), 139-174.
D. Fortunato and E. Jannelli, Infinitely many solutions for some nonlinear elliptic problems in symmetrical domains, Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 205-213.
J.G. Azorero and I. Peral, Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues, Comm. Partial Differential Equations 12 (1987), 1389-1430.
M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879-902.
K. Perera, Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal. 21 (2003), 301-309.
K. Perera, R. P. Agarwal and D. O'Regan, Morse theoretic aspects of p-Laplacian type operators, 161 Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2010.
K. Perera and A. Szulkin, p-Laplacian problems where the nonlinearity crosses an eigenvalue, Discrete Contin. Dyn. Syst. 13 (2005), 743-753.
M. Schechter and W. Zou, On the Brezis-Nirenberg problem, Arch. Ration. Mech. Anal. 197 (2010), 337-356.
Y. Wu and Y. Huang, Infinitely many sign-changing solutions for p-Laplacian equation involving the critical Sobolev exponent, Bound. Value Probl. 149, 10, 2013.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 9
