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
Mots clés : quasilinear elliptic equation, generic existence, variational inequality, rapidly growing operator

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