EXISTENCE RESULTS FOR A CLASS OF SEMILINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

Abstract

We consider the semilinear elliptic boundary value problem  = ∈∂Ω− Δ = ∈Ωu x xu x f u x x( ) 0;( ) λ ( ( ));where λ > 0 is a parameter, Ω is a bounded region in RN with a smooth boundary, and f is asmooth function. We prove, under some additional conditions, the existence of a positive solution for λlarge. We prove that our solution u for λ large is such that = →∞∈Ω|| u ||: sup| u(x) |xas λ →∞.Also, in the case of N = 1, we use a bifurcation theory to show that the solution is unstable.