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A nonsymmetric asymptotically linear elliptic problem. (English) Zbl 0844.35035

Let Ω be a bounded domain in N . The paper is concerned with the semilinear elliptic problem

Δu+g(x,u)=te 1 inΩ,u=0onΩ,(*)

where g(x,u)=αu + +βu - +g 0 (x,u), g 0 (x,u)/u0 as |u|, e 1 is the positive eigenvalue of the Laplacian and α,β,t. To (*) there corresponds a functional

f t (u)= Ω (1 2|u| 2 -G(x,u)+te 1 u)dx

in H 0 1 (Ω) and critical points of f t are solutions of (*). It is shown that for (α,β) in certain regions of 2 , if t is large enough, then (*) has at least one, two, three, respectively four solutions. Existence of one solution is shown by using a variant of the saddle point theorem of Rabinowitz. Two and three solutions are obtained by linking-type arguments where careful estimates are needed in order to show that certain linking levels are different. An additional argument gives a fourth critical point. It should also be noted that a rather general sufficient condition for f t to satisfy the Palais-Smale condition is given in this paper.


MSC:
35J65Nonlinear boundary value problems for linear elliptic equations
58E05Abstract critical point theory
35J20Second order elliptic equations, variational methods