CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 4, Number 2, Spring 1996

ASYMPTOTIC BEHAVIOR OF SOLUTION CONTINUA FOR

SEMILINEAR ELLIPTIC PROBLEMS

STANISLAUS MAIER-PAAPE AND KLAUS SCHMITT

ABSTRACT. We consider parameter dependent Dirich-let boundary value problems for semilinear elliptic equations whose nonlinear terms are linear a t infinity. Results are pre- sented about the asymptotic behavior of solution continua which bifurcate from infinity. In particular, we obtain the existence of infinitely many solutions in case the parameter equals a simple eigenvalue of the linearized problem.

1. Introduction. Let R be a bounded domain in Rn with a smooth boundary 8 0 , and let L be a self-adjoint elliptic operator. We consider the asymptotic behavior of solution continua of the nonlinear boundary value problem

with a real parameter X and a smooth function f : R + R which is asymptotically linear at infinity. This enables us to compare large normed solutions of (1.1) with solutions of the linear eigenvalue problem

We assume that @ is an eigensolution of (1.2) for a simple eigenvalue A@. It is known (cf. [g]) that in this case there is a bifurcation from infinity at Xo for the nonlinear problem, i.e., there exists a smooth branch of solutions (w(.; a),X(a)) in G'i1'(fi) x R, y E (0, l ) , for (1.1) ( a E R sufficiently large). The solutions converge to the eigensolution of the linear problem in the following sense:

Received by the editors on October 19, 1993, and in revised form on January 11, 1996.

Copyright 01996 Rocky Mountain Mathematics Consortium

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S. MAIER-PAAPE AND K. SCHMITT

and

(1.3) X(a) + Xo as a + co. In the case where is the principal eigenvalue of L = A (the Laplacian) Schaaf and Schmitt [7] have studied the behavior of this branch for a class of nonlinearities f(w) = w + g(w). They, in particular, discovered that for certain oscillatory nonlinearities, e.g., g(w) = sin(w), the branch bifurcating from infinity at A+ will oscillate about the hyperplane X = Xo infinitely often as a co whenever R is a convex domain and n = 1,2,3 and 4, whereas it will lie on one side of this hyperplane when n 2 5. The case where g(w) = sin(w) and related nonlinearities were also studied in detail by Dancer [2].

Thus the question arises whether oscillatory nonlinearities exist for which the bifurcating branch exhibits oscillatory behavior for all n E N. The aim of this article is to construct such a class of nonlinearities and study the exact asymptotic behavior of that branch, or, more generally, of that continuum. Aside from the following, no convexity assumption will be made on R as also it will not be assumed that Xo is the principal eigenvalue of L. Assuming that the eigensolution (T, has only smooth nodal surfaces which lie completely in the interior of R, we can construct a class of nonlinearities f such that for the associated nonlinear boundary problem X(a)-Xo has infinitely many sign changes. Hence, for these nonlinearities the problem (1.1) with X = Xo has infinitely many solutions (see Figure 1). We shall furthermore briefly discuss some cases, where our hypotheses on L, R and are satisfied. For instance, our results apply to all problems, where A+ is the principal eigenvalue. Moreover, all eigensolutions in dimension n = 1 and for n 2 2 radially symmetric solutions on ball domains for L = A can be handled as well. In the recent work of Kielhofer and Maier [4] a class of nonlinearities f is found which allow unbounded oscillatory continua of solutions in any dimension, again only for positive solutions. Here, however, the nonlinearities are not asymptotically linear, as was the case in the classical paper of Joseph and Lundgren [5], where (1.1) with f ( w ) = eW is considered on a ball.

2. O n solution continua. In the following we give the assumptions on L, R, the eigensolution (T,, and the nonlinearity f and prove some convergence rates for the solution continua.