Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

Abstract

Abstract. In this paper, we study bounded solutions of −∆u = f (u) on R n (where n = 2 and sometimes n = 3) and show that, for most f 's, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of − 2 ∆u = f (u) on Ω with Dirichlet or Neumann boundary conditions for small .The purpose of the present paper is to study the equationon R n (or a half space T ), where we are interested in solutions which are weakly stable or have finite Morse index (that is, have only finitely many negative eigenvalues in some suitable generalized sense). Usually, we study positive solutions (but not always). For n = 2 and sometimes for n = 3 we prove that these solutions are very simple and easy to understand. (For example, if n = 2, the weakly stable solutions usually have to be constant.) This is an interesting contrast to the results in [8] where for certain nonlinearities there are a great many positive bounded solutions which are periodic in some variables and decay in others. In addition, for many f , it is easy to use variational methods (or bifurcation methods) to construct many solutions periodic (and non-constant) in all variables. Thus it seems that the structure of all solutions of (1) may be quite complicated. Our results show that the finite Morse index solutions are usually quite simple. We also prove closely related results on half spaces.As an application of these ideas, we have a number of results on the solutions (usually but not always positive) ofwhere Ω is a bounded open set in R 2 (sometimes in R 3 ) and we have homogeneous Neumann or Dirichlet boundary conditions. For many f 's we obtain the exact number of stable positive solutions for small positive and show it is independent of the shape of the domain Ω (and easy to calculate). This contrasts strongly with the results for all positive solutions in [9], [10] and [11] for small positive and results for the stable positive solutions when is not small. We also prove that stable positive solutions are constant in the case of Neumann boundary conditions, and for either boundary condition there are no stable sign-changing positive solutions for small positive (for many f 's). These two results contrast with results in [31] and [16].

Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

“…Theorem 4 recovers and improves upon a result of [26], which held for N 5 (see also [2,3,8,11,12,18,19,21,[23][24][25] for some other results concerning problem (4). )…”

“…Proof of Theorem 5. By proceeding as in the first part of the proof of Theorem 2 in [12] we get that the even extension of u to R N is a bounded below stable solution of (1) in R N , N 10. This function must be constant by Theorem 1.…”

The work of Norman Dancer

“…(See [51] for an exampe of a different nature.) There is a converse conclusion to Theorem 8.5, namely, if f (M ) = 0 for some M > 0, then for each c ∈ (0, M ) satisfying (8.3), problem (8.2) has a weakly stable positive solution u for all small > 0 having the properties described in Theorem 8.5; moreover, the number of weakly stable positive solutions of (8.2) with small > 0 that have L ∞ -norm smaller than M is the same as the number of constants c ∈ (0, M ) satisfying (8.3) ( [42,43]).…”

Dynamics of a reaction-diffusion system of autocatalytic chemical reaction

“…It is possible to extend the results here for unit ball to a general smooth domain Ω with small diffusion coefficient D. The key would be establishing corresponding exact multiplicity of nonnegative steady states. Ideas in [4,5] can be adapted to achieve that, but the details will appear elsewhere.…”