# 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$\phantom{\rule{-11.38109pt}{0ex}}\begin{array}{c}-\mathrm{div}\left(a\left(|\nabla u|\right)\nabla u\right)=\lambda g\left(x,u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{on}\phantom{\rule{4.0pt}{0ex}}\partial \Omega ,\hfill \end{array}$in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all $\lambda >0$ 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 (2004): 677-691. <http://eudml.org/doc/244618>.

@article{Le2004,
abstract = {We consider the eigenvalue problem$\hspace*\{-11.38109pt\} \begin\{array\}\{l\} -\{\rm div\} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox\{ in \} \Omega \\ u = 0 \;\mbox\{ on \} \partial \Omega , \end\{array\}$in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all $\lambda &gt;0$ 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},
language = {eng},
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/244618},
volume = {10},
year = {2004},
}

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
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 4
SP - 677
EP - 691
AB - We consider the eigenvalue problem$\hspace*{-11.38109pt} \begin{array}{l} -{\rm div} (a(|\nabla u |)\nabla u) = \lambda g(x, u) \;\mbox{ in } \Omega \\ u = 0 \;\mbox{ on } \partial \Omega , \end{array}$in the case where the principal operator has rapid growth. By using a variational approach, we show that under certain conditions, almost all $\lambda &gt;0$ are eigenvalues.
LA - eng
KW - quasilinear elliptic equation; generic existence; variational inequality; rapidly growing operator
UR - http://eudml.org/doc/244618
ER -

## References

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1. [1] R. Adams, Sobolev spaces. Academic Press, New York (1975). Zbl0314.46030MR450957
2. [2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381. Zbl0273.49063MR370183
3. [3] K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981) 102-129. Zbl0487.49027MR614246
4. [4] F.H. Clarke, Optimization and nonsmooth analysis. SIAM, Philadelphia (1990). Zbl0696.49002MR1058436
5. [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. Zbl0959.35057MR1777463
6. [6] T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces. J. Diff. Equations 10 (1971) 507-528. Zbl0218.35028MR298472
7. [7] T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal. 8 (1971) 52-75. Zbl0216.15702MR301500
8. [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. Zbl0936.35067MR1694787
9. [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. Zbl0239.35045MR342854
10. [10] J.P. Gossez and R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces. Proc. Roy. Soc. Edinb. A 132 (2002) 891-909. Zbl1014.35071MR1926921
11. [11] J.P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces. Nonlinear Anal. 11 (1987) 379-392. Zbl0643.49006MR881725
12. [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. A 129 (1999) 787-809. Zbl0935.35044MR1718530
13. [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. Zbl0996.47052MR1650239
14. [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. Zbl0964.35055MR1775181
15. [15] M.A. Krasnosels’kii and J. Rutic’kii, Convex functions and Orlicz spaces. Noorhoff, Groningen (1961).
16. [16] A. Kufner, O. John and S. Fučic, Function spaces. Noordhoff, Leyden (1977). Zbl0364.46022
17. [17] V.K. Le, A global bifurcation result for quasilinear eliptic equations in Orlicz-Sobolev space. Topol. Methods Nonlinear Anal. 15 (2000) 301-327. Zbl0971.35029MR1784144
18. [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. Zbl1034.35056MR1895569
19. [19] V.K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients. J. London Math. Soc. 62 (2000) 852-872. Zbl1013.35032MR1794290
20. [20] V. Mustonen and M. Tienari, An eigenvalue problem for generalized Laplacian in Orlicz-Sobolev spaces. Proc. Roy. Soc. Edinb. A 129 (1999) 153-163. Zbl0926.46030MR1669197
21. [21] V. Mustonen, Remarks on inhomogeneous elliptic eigenvalue problems. Part. Differ. Equ. Lect. Notes Pure Appl. Math. 229 (2002) 259-265. Zbl1142.35538MR1913336
22. [22] Z. Naniewicz and P.D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New York (1995). Zbl0968.49008MR1304257
23. [23] P. Rabinowitz, Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973) 162-202. Zbl0255.47069MR320850
24. [24] M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Brasil Mat. 20 (1990) 49-58. Zbl0719.58032MR1143173
25. [25] M. Struwe, Variational methods. 2nd ed., Springer, Berlin (1991). Zbl0746.49010
26. [26] M. Tienari, Ljusternik-Schnirelmann theorem for the generalized Laplacian. J. Differ. Equations 161 (2000) 174-190. Zbl0946.35057MR1740361

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