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

Le, Vy Khoi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 4, pp. 677-691

We consider the eigenvalue problem

$$\begin{array}{c}-\mathrm{div}\left(a\right(\left|\nabla u\right|\left)\nabla u\right)=\lambda g(x,u)\text{in}\Omega \hfill \\ u=0\text{on}\partial \Omega ,\hfill \end{array}$$

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.

Classification : 35J65, 35J20, 35J60, 47J30, 49J40, 58E05
Keywords: quasilinear elliptic equation, generic existence, variational inequality, rapidly growing operator

[1] R. Adams, Sobolev spaces. Academic Press, New York (1975). | Zbl | MR

[2] A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973) 349-381. | Zbl | MR

[3] K.C. Chang, Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80 (1981) 102-129. | Zbl | MR

[4] F.H. Clarke, Optimization and nonsmooth analysis. SIAM, Philadelphia (1990). | Zbl | MR

[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. | Zbl | MR

[6] T. Donaldson, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces. J. Diff. Equations 10 (1971) 507-528. | Zbl | MR

[7] T. Donaldson and N. Trudinger, Orlicz-Sobolev spaces and imbedding theorems. J. Funct. Anal. 8 (1971) 52-75. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[11] J.P. Gossez and V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces. Nonlinear Anal. 11 (1987) 379-392. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[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). | Zbl

[17] V.K. Le, A global bifurcation result for quasilinear eliptic equations in Orlicz-Sobolev space. Topol. Methods Nonlinear Anal. 15 (2000) 301-327. | Zbl | MR

[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. | Zbl | MR

[19] V.K. Le and K. Schmitt, Quasilinear elliptic equations and inequalities with rapidly growing coefficients. J. London Math. Soc. 62 (2000) 852-872. | Zbl | MR

[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. | Zbl | MR

[21] V. Mustonen, Remarks on inhomogeneous elliptic eigenvalue problems. Part. Differ. Equ. Lect. Notes Pure Appl. Math. 229 (2002) 259-265. | Zbl | MR

[22] Z. Naniewicz and P.D. Panagiotopoulos, Mathematical theory of hemivariational inequalities and applications. Marcel Dekker, New York (1995). | Zbl | MR

[23] P. Rabinowitz, Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973) 162-202. | Zbl | MR

[24] M. Struwe, Existence of periodic solutions of Hamiltonian systems on almost every energy surface. Bol. Soc. Brasil Mat. 20 (1990) 49-58. | Zbl | MR

[25] M. Struwe, Variational methods. 2nd ed., Springer, Berlin (1991). | Zbl

[26] M. Tienari, Ljusternik-Schnirelmann theorem for the generalized Laplacian. J. Differ. Equations 161 (2000) 174-190. | Zbl | MR