# Generic existence result for an eigenvalue problem with rapidly growing principal operator

• Volume: 10, Issue: 4, page 677-691
• ISSN: 1292-8119

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## Abstract

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We consider the eigenvalue problem $\begin{array}{c}-\mathrm{div}\left(a\left(|\nabla u|\right)\nabla u\right)=\lambda g\left(x,u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega u=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}\partial \Omega ,\hfill \end{array}$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.

## How to cite

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Le, Vy Khoi. "Generic existence result for an eigenvalue problem with rapidly growing principal operator." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 677-691. <http://eudml.org/doc/90751>.

@article{Le2010,
abstract = { We consider the eigenvalue problem $$\begin\{array\}\{l\} \displaystyle-\{\rm div\} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox\{ in \} \Omega u = 0 \;\mbox\{ on \} \partial\Omega , \end\{array\}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues. },
author = {Le, Vy Khoi},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Quasilinear elliptic equation; generic existence; variational inequality; rapidly growing operator.; quasilinear elliptic equation; rapidly growing operator},
language = {eng},
month = {3},
number = {4},
pages = {677-691},
publisher = {EDP Sciences},
title = {Generic existence result for an eigenvalue problem with rapidly growing principal operator},
url = {http://eudml.org/doc/90751},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Le, Vy Khoi
TI - Generic existence result for an eigenvalue problem with rapidly growing principal operator
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 677
EP - 691
AB - We consider the eigenvalue problem $$\begin{array}{l} \displaystyle-{\rm div} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox{ in } \Omega u = 0 \;\mbox{ on } \partial\Omega , \end{array}$$ in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all λ > 0 are eigenvalues.
LA - eng
KW - Quasilinear elliptic equation; generic existence; variational inequality; rapidly growing operator.; quasilinear elliptic equation; rapidly growing operator
UR - http://eudml.org/doc/90751
ER -

## References

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2. A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal.14 (1973) 349-381.  Zbl0273.49063
3. K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl.80 (1981) 102-129.  Zbl0487.49027
4. F.H. Clarke, Optimization and nonsmooth analysis. SIAM, Philadelphia (1990).  Zbl0696.49002
5. P. Clément, M. García-Huidobro, R. Manásevich and K. Schmitt, Mountain pass type solutions for quasilinear elliptic equations. Calc. Var.11 (2000) 33-62.  Zbl0959.35057
6. T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces. J. Diff. Equations10 (1971) 507-528.  Zbl0207.41501
7. T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal.8 (1971) 52-75.  Zbl0216.15702
8. M. García-Huidobro, V.K. Le, R. Manásevich and K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: An Orlicz-Sobolev space setting. Nonlinear Diff. Eq. Appl.6 (1999) 207-225.  Zbl0936.35067
9. J.P. Gossez, Nonlinear elliptic boundary value problems for equations with rapidly or slowly increasing coefficients. Trans. Amer. Math. Soc.190 (1974) 163-205.  Zbl0239.35045
10. J.P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces. Proc. Roy. Soc. Edinb. A132 (2002) 891-909.  Zbl1014.35071
11. J.P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces. Nonlinear Anal.11 (1987) 379-392.  Zbl0643.49006
12. L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${𝐑}^{N}$. Proc. Roy. Soc. Edinb. A129 (1999) 787-809.  Zbl0935.35044
13. L. Jeanjean and J.F. Toland, Bounded Palais-Smale mountain-pass sequences. C.R. Acad. Sci. Paris Ser. I Math.327 (1998) 23-28.  Zbl0996.47052
14. N.C. Kourogenis and N.S. Papageorgiou, Nonsmooth critical point theory and nonlinear elliptic equations at resonance. J. Austral. Math. Soc. (Ser. A)69 (2000) 245-271.  Zbl0964.35055
15. M.A. Krasnosels'kii and J. Rutic'kii, Convex functions and Orlicz spaces. Noorhoff, Groningen (1961).
16. A. Kufner, O. John and S. Fučic, Function spaces. Noordhoff, Leyden (1977).
17. V.K. Le, A global bifurcation result for quasilinear eliptic equations in Orlicz-Sobolev space. Topol. Methods Nonlinear Anal.15 (2000) 301-327.  Zbl0971.35029
18. V.K. Le, Nontrivial solutions of mountain pass type of quasilinear equations with slowly growing principal parts. J. Diff. Int. Eq.15 (2002) 839-862.  Zbl1034.35056
19. V.K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients. J. London Math. Soc.62 (2000) 852-872.  Zbl1013.35032
20. V. Mustonen and M. Tienari, An eigenvalue problem for generalized Laplacian in Orlicz-Sobolev spaces. Proc. Roy. Soc. Edinb. A129 (1999) 153-163.  Zbl0926.46030
21. V. Mustonen, Remarks on inhomogeneous elliptic eigenvalue problems. Part. Differ. Equ. Lect. Notes Pure Appl. Math.229 (2002) 259-265.  Zbl1142.35538
22. Z. Naniewicz and P.D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New York (1995).  Zbl0968.49008
23. P. Rabinowitz, Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math.3 (1973) 162-202.  Zbl0255.47069
24. M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Brasil Mat.20 (1990) 49-58.  Zbl0719.58032
25. M. Struwe, Variational methods. 2nd ed., Springer, Berlin (1991).
26. M. Tienari, Ljusternik-Schnirelmann theorem for the generalized Laplacian. J. Differ. Equations161 (2000) 174-190.  Zbl0946.35057

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