Generic properties of the Rabinowitz unbounded continuum

Daniele Bartolucci; Yeyao Hu; Aleks Jevnikar; Wen Yang

doi:10.1515/ans-2022-0062

Article Open Access

Generic properties of the Rabinowitz unbounded continuum

Daniele Bartolucci , Yeyao Hu , Aleks Jevnikar and Wen Yang

Published/Copyright: April 15, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advanced Nonlinear Studies Volume 23 Issue 1

Abstract

In this article, we prove that, generically in the sense of domain variations, any solution to a nonlinear eigenvalue problem is either nondegenerate or the Crandall-Rabinowitz transversality condition that is satisfied. We then deduce that, generically, the unbounded Rabinowitz continuum of solutions is a simple analytic curve. The global bifurcation diagram resembles the classic model case of the Gel’fand problem in two dimensions.

Keywords: Rabinowitz continuum; bifurcation analysis; generic properties

MSC 2010: 35B30; 35B32; 35J61

1 Introduction

Let Ω ⊂ R N be an open and bounded domain of class C 4 , we are concerned with generic properties of the Rabinowitz [22] unbounded continuum of C 0 2 , r ( Ω ¯ ) -solutions of

(1.1) − Δ v = μ f ( v ) in Ω v = 0 on ∂ Ω ,

with μ ≥ 0 and f satisfying:

(H1) f : ( a , + ∞ ) → ( 0 , + ∞ ) of class C 2 for some a < 0 , f ′ ( t ) > 0 , f ″ ( t ) > 0 , ∀ t ∈ ( a , + ∞ ) .

In particular, by the maximum principle, we have v > 0 in Ω . It follows from [22] that there exists a closed (in the [ 0 , + ∞ ) × C 0 2 , r ( Ω ¯ ) -topology) connected and unbounded set of solutions ( μ , v μ ) of (1.1), which we denote by ℛ ∞ that contains the unique solution for μ = 0 , which is ( μ , v μ ) = ( 0 , 0 ) . Of course, it is not true in general that ℛ ∞ is a simple curve with no bifurcation points, see, for example, [18,19,21]. If f is real analytic, then ℛ ∞ is also a path-connected set [6]. On the other side, much more is known for certain classes of nonlinearities in the radial case (see [9] and in particular [7,14,15] for a review) or, limited to Ω ⊂ R 2 , for symmetric and convex geometries (see [10]). Actually, in these cases in particular, ℛ ∞ is a one-dimensional connected manifold in [ 0 , + ∞ ) × C 0 2 , r ( Ω ¯ ) whose boundary is ( 0 , 0 ) . See also [1] for a more detailed description of the qualitative ̲ behavior of ℛ ∞ for f ( t ) = e t and N = 2 .

Our aim here is to initiate the qualitative study (in the spirit of [1]) of ℛ ∞ in a general setting. To this end, we first prove that, for “almost any domain” in a suitably defined sense, a solution ( μ , u μ ) of (1.1) is either nondegenerate or the classical Crandall and Rabinowitz [4] transversality condition that is satisfied. We then deduce that, generically with respect to domain variations, under suitable regularity and growth condition on f , ℛ ∞ is indeed a one-dimensional connected manifold in [ 0 , + ∞ ) × C 0 2 , r ( Ω ¯ ) whose boundary is ( 0 , 0 ) and that asymptotically approaches the vertical axis, as shown in some particular cases in the classical result of [16]. Although this result seems to be well-known, we could not find a statement of this sort in the literature.

As far as we are concerned with generic properties with respect to domain variations, many results are by now classical, see, for example, [11,24], and references quoted therein. Among many other applications which we cannot discuss here, it follows from [11,24] that, for a fixed μ , then for “almost any domain” any solution of (1.1) is nondegenerate. This is obviously false in general if μ is not fixed ([4]). Actually, these sort of results hold for much more general semilinear elliptic partial differential equations (PDEs) and are used [11,24] to infer that the number of solutions of certain equations is, generically with respect to domain variations, either finite or at most countable.

Generic simpleness of eigenvalues and/or nondegeneracy properties with respect to variations of μ and/or coefficients of the equations are also well-known, starting with [12,17,23,24,26] as later improved in the real-analytic framework in [5] (see also [2,3]).

On the other side, results of the sort considered in this article have been obtained in [23]. First of all, as mentioned above, one can find in [23] a detailed discussion of the fact that, in a generic sense with respect to variations in coefficients, and under suitable regularity assumptions, the set of solutions of a quasilinear elliptic equation with f ( v ) = d ( x ) v + o ( ‖ v ‖ ) as ‖ v ‖ → 0 is the union of at most countably many curves of class C k − 1 for some k ≥ 3 . Also, a short discussion of the genericity with respect to domain variations is provided in [23], claiming that coupling some arguments in [24] and those of [23] yields the same result, still for f ( v ) = d ( x ) v + o ( ‖ v ‖ ) as ‖ v ‖ → 0 .

Compared to [23], our results contain some relevant differences, which we shortly describe hereafter.

First, the nonlinearity considered in [23] has essentially the form of f ( v ) = d ( x ) v + o ( ‖ v ‖ ) and hence, in particular, the branches analyzed therein are those bifurcating from the line of trivial solutions. On the contrary, we are interested to the qualitative properties of the branch of solutions bifurcating from ( 0 , 0 ) with f superlinear (see (H1)).

Second and more importantly, we attack the problem with a different approach. Indeed, we think this could be a first promising step toward the understanding of the qualitative behavior of the solution curves (in the same spirit of [1]). Thus, we do not argue as in [23], where the fact that the set of solutions is the union of at most countably many regular curves follows at once by the argument in ([24], Section 4). Instead, as mentioned above, we first prove a result of independent interest, Theorem 1.1, showing that, in a generic sense with respect to domain variations, any solution on the continuum ̲ is either nondegenerate or the Crandall-Rabinowitz transversality condition is satisfied. Next, by the well-known bending arguments based on the Crandall-Rabinowitz transversality condition, we deduce by the analytic implicit function theorem [3] that around a singular point, the curve of solutions is real analytic and has no bifurcation points. We then apply this result to the qualitative study of ℛ ∞ . In particular, in view of the assumption (H2), we see that the branch asymptotically approaches the vertical axis. Of course, it is likely that analyticity of the solution curves considered in [23] could follow as well from the arguments in [23] under suitable assumptions.

The argument here works essentially as in [23,24], although we rely on the more recent reference [11], where one can find a detailed and self-contained exposition of the theory of domain variations (see in particular Chapter 2 in [11]).

For Ω 0 , a bounded domain of class C 4 (see Section 3 for details), we denote by Diff 4 ( Ω 0 ) the set of diffeomorphisms h : Ω 0 ¯ → Ω ¯ of class C 4 . We recall that a subset of a metric space is said to be:

nowhere dense, if its closure has empty interior;
meager (or of first Baire category), if it is the union of countably many nowhere dense sets.

Once more, it is likely that this result is known to experts in the field, still we could not find a statement of this sort in the literature.

Here, L μ is the linearized operator relative to (1.1) (see Section 2). Then, we have,

Theorem 1.1

Let f : ( a , + ∞ ) → ( 0 , + ∞ ) be of class C 2 for some a < 0 . For any Ω 0 ⊂ R N of class C 4 , there exists a meager set ℱ ⊂ Diff 4 ( Ω 0 ) , depending on f , N , and Ω 0 , such that if h ∈ Diff 4 ( Ω 0 ) ⧹ ℱ then, for any solution ( μ , v μ ) of (1.1) on Ω ≔ h ( Ω 0 ) with μ > 0 , it holds: either

Ker ( L μ ) = ∅ , or
Ker ( L μ ) = span { ϕ } is one-dimensional and ∫ Ω f ( v μ ) ϕ ≠ 0 .

In particular, we deduce the following result about the Rabinowitz [22] unbounded continuum of solutions of (1.1) for f real analytic, which satisfies (H1) and,

(H2) for any δ > 0 , there exists C δ > 0 (depending also by f , N , and Ω ) such that v μ ≤ C δ for any solution of (1.1) with μ ≥ δ .

It is well-known that, for Ω a bounded domain of class C 4 , (H2) is satisfied under suitable growth assumptions on f , as for example those in [8] (here F ( t ) = ∫ 0 t f ( s ) d s ),

lim t → + ∞ f ( t ) t = + ∞ , lim t → + ∞ f ( t ) t β = 0 , β = N + 2 N − 2 if N ≥ 3 , β < + ∞ if N = 2 , limsup t → + ∞ t f ( t ) − θ F ( t ) t 2 f 2 N ( t ) ≤ 0 , for some θ ∈ 0 , 2 N N − 2 , and if N ≥ 3 , f ( t ) t − N + 2 N − 2 is nonincreasing in ( 0 , + ∞ ) .

Clearly, the model nonlinearities f ( t ) = ( 1 + t ) p , 1 < p < N + 2 N − 2 , N ≥ 3 , p > 1 , N = 2 , fit these assumptions. However, there are many other cases where ( H2 ) is satisfied, as for example f ( t ) = e t , N = 2 ([20]). Then, we have:

Theorem 1.2

Let f be real analytic and satisfying (H1) and (H2), Ω 0 ⊂ R N of class C 4 , ℱ ⊂ Diff 4 ( Ω 0 ) as defined by Theorem 1.1 and pick h ∈ Diff 4 ( Ω 0 ) ⧹ ℱ . If Ω = h ( Ω 0 ) , then the Rabinowitz unbounded continuum ℛ ∞ of solution of (1.1) is a one-dimensional real analytic manifold with boundary ( μ ( 0 ) , v ( 0 ) ) = ( 0 , 0 ) . In particular,

ℛ ∞ = { [ 0 , ∞ ) ∋ s ↦ ( μ ( s ) , v ( s ) ) ∈ [ 0 , + ∞ ) × C 0 2 , r ( Ω ¯ ) } ,

is a continuous simple curve without bifurcation points where ( μ ( 0 ) , v ( 0 ) ) = ( 0 , 0 ) and μ ( s ) → 0 + and ‖ v ( s ) ‖ ∞ → + ∞ as s → ∞ .

Remark 1.3

Theorem 1.1 can be generalized to the case of uniformly elliptic operators such as L u ≔ div ( A ( ∇ u ) ) + b → ⋅ ∇ u + c u , where A = ( a i j ) , a i j ( x ) , b j ( x ) , and c ( x ) are smooth up to the boundary. Therefore, the generic bending result (Theorem 1.2) also follows if one replaces the Laplace operator in (1.1) by uniformly elliptic operators.

2 Well-known results

Let X = R × C 0 2 , r ( Ω ¯ ) , we introduce the map,

(2.1) F : X → C r ( Ω ¯ ) , F ( μ , v ) ≔ − Δ v − μ f ( v ) ,

and its differential with respect to ( μ , v ) , that is, the linear operator,

D μ , v F ( μ , v ) : X → C r ( Ω ¯ ) ,

which acts as follows:

D μ , v F ( μ , v ) [ μ ˙ , v ˙ ] = D v F ( μ , v ) [ v ˙ ] + d μ F ( μ , v ) [ μ ˙ ] ,

where we have introduced the differential operators,

D v F ( μ , v ) [ v ˙ ] = − Δ v ˙ − μ f ′ ( v ) v ˙ , v ˙ ∈ C 0 2 , r ( Ω ¯ ) ,

d μ F ( μ , v ) [ μ ˙ ] = − f ( v ) μ ˙ , μ ˙ ∈ R .

For a fixed solution ( μ , v μ ) , the eigenvalues of L μ ≔ D v F ( μ , v μ ) form an increasing sequence and are denoted by σ k , k ∈ N , which satisfy:

L μ ϕ = σ k ϕ , ϕ ∈ C 0 2 , r ( Ω ¯ ) .

By the Fredholm alternative, the implicit function theorem applies around any solution of (1.1) as follows:

Lemma 2.1

Let ( μ 0 , v 0 ) be a solution of (1.1) with μ = μ 0 ≥ 0 .

If 0 is not an eigenvalue of L μ 0 , then:

L μ 0 is an isomorphism;
There exists an open neighborhood J ⊂ R of μ 0 and ℬ ⊂ C 0 2 , r ( Ω ¯ ) of v 0 , such that the set of solutions of (1.1) in J × ℬ is a curve of class C 2 , J ∋ μ ↦ v μ ∈ ℬ .

Next, we state the well-known bending result of [4] for solutions of (1.1) just with an additional observation about the case where f is real analytic in ( a , + ∞ ) for some a < 0 . The conclusions deduced in this particular case are straightforward consequences of general and well-known facts of analytic bifurcation theory [3].

Proposition 2.2

[4] Let ( μ , v μ ) be a solution of (1.1) with μ > 0 and suppose that the kth eigenvalue of L μ satisfies σ k = 0 and is simple, that is, it admits only one eigenfunction, ϕ k ∈ C 0 2 , r ( Ω ¯ ) . If

∫ Ω f ( v μ ) ϕ k ≠ 0 ,

then there exists ε > 0 , an open neighborhood U of ( μ , v μ ) in X and a curve ( − ε , ε ) ∋ s ↦ ( μ ( s ) , v ( s ) ) of class C 2 such that ( μ ( 0 ) , v ( 0 ) ) = ( μ , v μ ) and the set of solutions of (1.1) in U has the form ( μ ( s ) , v ( s ) ) with, v ( s ) = v μ + s ϕ k + ξ ( s ) , and

∫ Ω f ( v ( s ) ) ξ ( s ) ϕ k = 0 , s ∈ ( − ε , ε ) .

Moreover, it holds

(2.2) ξ ( 0 ) ≡ 0 ≡ ξ ′ ( 0 ) , μ ′ ( 0 ) = 0 ,

and there exists a continuous curve ( σ ( s ) , ϕ ( s ) ) , such that L μ ( s ) ϕ ( s ) = σ ( s ) ϕ ( s ) , s ∈ ( − ε , ε ) , ϕ ( 0 ) = ϕ k , σ ( 0 ) = σ k , and

σ ( s ) ∫ Ω f ′ ( v ( s ) ) ϕ ( s ) v ( s ) a n d μ ′ ( s ) ∫ Ω f ( v ( s ) ) ϕ ( s )

have the same zeroes and, whenever μ ′ ( s ) ≠ 0 , the same sign. In particular,

σ ( s ) μ ′ ( s ) = ∫ Ω f ( v μ ) ϕ k + o ( 1 ) ∫ Ω ϕ k 2 + o ( 1 ) , a s s → 0 .

If f is real analytic in ( a , + ∞ ) for some a < 0 , then μ ( s ) , v ( s ) , σ ( s ) , and ϕ ( s ) are the real analytic functions of s ∈ ( − ε , ε ) and in particular either μ ( s ) is constant in ( − ε , ε ) or μ ′ ( s ) ≠ 0 , σ ( s ) ≠ 0 in ( − ε , ε ) ⧹ { 0 } and σ ( s ) is simple in ( − ε , ε ) .

3 Generic properties of the Rabinowitz continuum

In this section, we prove Theorems 1.1 and 1.2.

Let us recall few definitions and set some notations first.

Definition 3.1

A domain Ω is of class C k ( C k , r ) , k ≥ 1 , if for each x 0 ∈ ∂ Ω , there exists a ball B = B r ( x 0 ) and a one-to-one map Θ : B ↦ U ⊂ R N such that Θ ∈ C k ( B ) ( C k , r ( B ) ) , Θ − 1 ∈ C k ( U ) ( C k , r ( U ) ) and the following holds:

Θ ( Ω ∩ B ) ⊂ R + N and Θ ( ∂ Ω ∩ B ) ⊂ ∂ R + N .

It is well-known (see, for example, [11]) that this is equivalent to say that there exists r > 0 and M > 0 such that, given any ball B ⊂ R N , then, after suitable rotation and translations, it holds:

Ω ∩ B = { ( x 1 , x … , x N ) : x N < f ( x 1 , … , x N − 1 ) } ∩ B

and

∂ Ω ∩ B = { ( x 1 , x … , x N ) : x N = f ( x 1 , … , x N − 1 ) } ∩ B ,

for some f ∈ C k ( R N − 1 ) ( C k , r ( R ) ) whose norm is not larger than M .

Definition 3.2

Let Ω ⊂ R N be an open and bounded domain of class C m , m ≥ 1 . C m ( Ω ¯ ; R N ) is the Banach space of continuous and m -times differentiable maps on Ω , whose derivatives of order j = 0 , 1 , … , m extend continuously on Ω ¯ . Diff m ( Ω ) ⊂ C m ( Ω ¯ ; R N ) is the open subset of C m ( Ω ¯ ; R N ) whose elements are C m imbeddings on Ω ¯ , that is, of maps h : Ω ¯ ↦ R N , which are diffeomorphisms of class C m on their images h ( Ω ¯ ) .

We recall that if X and Z are Banach spaces and T : X → Z is linear and continuous, then T is Fredholm (semi-Fredholm) if R ( T ) (the range of T ) is closed and both dim(Ker ( T ) ) and codim ( R ( T ) ) are finite. If T is Fredholm, then the index of T is

ind ( T ) = dim ( Ker ( T ) ) − codim ( R ( T ) ) .

We refer to [13] for further details about Fredholm operators. Given a Banach space X and x ∈ X , we will denote by T x X the tangent space at x .

Definition 3.3

Let X and Z be Banach spaces, A ⊂ X an open set, and F : A → Z a C 1 map. Suppose that for any x ∈ A , the Fréchet derivative D x F ( x ) : T x X → T η Z is a Fredholm operator. A point x ∈ A is a regular point ̲ if D x F ( x ) is surjective, is a singular point ̲ otherwise. The image of a singular point η = F ( x ) ∈ Z is a singular value ̲ . The complement of the set of singular values in Z is the set of regular values ̲ .

The following theorem is a particular case of a more general transversality result proved in [11], see also [25].

Theorem 3.4

[11] Let X , ℋ , and Z be separable Banach spaces, A ⊆ X × ℋ an open set, Φ : A → Z a map of class C k , and η ∈ Z .

Suppose that for each ( x , h ) ∈ Φ − 1 ( η ) it holds:

D x Φ ( x , h ) : T x X → T η Z is a Fredholm operator with index < k ;
D Φ ( x , h ) = ( D x Φ ( x , h ) , D h Φ ( x , h ) ) : T x X × T h ℋ → T η Z is surjective.

Let A h = { x : ( x , h ) ∈ A } and

ℋ crit = { h : η i s a s i n g u l a r v a l u e o f Φ ( ⋅ , h ) : A h → Z } .

Then, ℋ crit is meager in ℋ .

We are ready to present the proof of Theorem 1.1.

Proof of Theorem 1.1

Let Ω 0 be as in the statement and let us define

X Ω 0 = R × C 0 2 , r ( Ω 0 ¯ ) .

We define the maps,

F Ω 0 : X Ω 0 → C r ( Ω 0 ¯ ) , F Ω 0 ( μ , v ) = Δ v + μ f ( v ) .

Next, for fixed h ∈ Diff 4 ( Ω 0 ) and v ∈ C 0 2 , r ( h ( Ω 0 ) ¯ ) , we define the pull-back,

h ∗ ( v ) ( x ) = v ( h ( x ) ) , x ∈ Ω 0 ¯ .

Clearly, h ∗ is an isomorphism of C 0 2 , r ( h ( Ω 0 ) ¯ ) onto C 0 2 , r ( Ω 0 ¯ ) with inverse h ∗ − 1 = ( h − 1 ) ∗ . For any such h , it is well defined the map

F h ( Ω 0 ) : X h ( Ω 0 ) → C r ( h ( Ω 0 ) ¯ )

and then we can set

h ∗ F h ( Ω 0 ) h ∗ − 1 : X Ω 0 × Diff 4 ( Ω 0 ) → C r ( Ω 0 ¯ ) .

Putting ℋ = Diff 4 ( Ω 0 ) , η = 0 ∈ Z = C r ( Ω 0 ¯ ) , we will apply Theorem 3.4 to the map Φ = Φ ( μ , v , h ) defined as follows:

Φ : A → R × C r ( Ω 0 ¯ ) , A = X Ω 0 × ℋ ,

Φ ( μ , v , h ) = h ∗ F h ( Ω 0 ) h ∗ − 1 ( μ , v ) .

Step 1: Our aim is to show that assumptions ( i ) and ( i i ) of Theorem 3.4 hold.

As in [11], it is very useful for the discussion to denote by ( μ ˙ , v ˙ , h ˙ ) ∈ R × C 0 2 , r ( Ω 0 ¯ ) × C 4 ( Ω 0 ¯ ; R N ) the elements of the tangent space at points ( μ , v , h ) ∈ X Ω 0 × ℋ .

First of all observe that for fixed h ∈ Diff 4 ( Ω 0 ) , the linearized operator,

D μ , v Φ ( μ , v , h ) : R × C 0 2 , r ( Ω 0 ¯ ) → C r ( Ω 0 ¯ ) ,

acts as follows on ( μ ˙ , v ˙ ) ∈ R × C 0 2 , r ( Ω 0 ¯ ) ,

D μ , v Φ ( μ , v , h ) [ μ ˙ , v ˙ ] = h ∗ ( Δ v ˙ ∗ + μ f ′ ( v ∗ ) v ˙ ∗ + f ( v ∗ ) μ ˙ ) ,

where

v ∗ = ( h ∗ ) − 1 v , v ˙ ∗ = ( h ∗ ) − 1 v ˙ .

Since any diffeomorphism of class C 4 maps the Laplace operator to a uniformly elliptic operator with C 2 coefficients, by standard elliptic estimates, it is not difficult to see that D μ , v Φ ( μ , v , h ) is a Fredholm operator of index 1.

This fact proves ( i ) whenever we can show that Φ ∈ C k ( A ) for some k ≥ 2 . The regularity of Φ with respect to h is the same as that of F h ( Ω 0 ) with respect to v , see chapter 2 in [11]. Therefore, we have Φ ∈ C 3 ( A ) , as claimed.

Next, we prove ( i i ) , that is, we show that η = 0 is a regular value for the map ( μ , v , h ) → Φ ( μ , v , h ) . We argue by contradiction and suppose that there exists a singular point ( μ ¯ , v ¯ , h ¯ ) of Φ such that Φ ( μ ¯ , v ¯ , h ¯ ) = 0 .

First of all, let us define Ω = h ¯ ( Ω 0 ) , u ¯ = ( h ¯ ∗ ) − 1 v ¯ ∈ C 0 2 , r ( Ω ¯ ) , and Φ ^ ( μ , u , h ) on X Ω × Diff 4 ( Ω ) as follows:

Φ ^ ( μ , u , h ) = h ∗ F Ω h ∗ − 1 ( μ , u ) ,

where

F Ω : X Ω → C r ( Ω ¯ ) , F Ω ( μ , u ) = Δ u + μ f ( u ) .

Let i Ω ∈ Diff 4 ( Ω ) be the identity map. By construction, in these new coordinates, the map Φ ^ ( μ , u , h ) has a singular point ( μ ¯ , u ¯ , i Ω ) such that Φ ^ ( μ ¯ , u ¯ , i Ω ) = 0 , that is, by assumption the derivative D μ , u , h Φ ^ ( μ ¯ , u ¯ , i Ω ) is not surjective. Putting

f ¯ = f ( u ¯ ) , f ¯ ′ = f ′ ( u ¯ ) ,

a subtle evaluation shows that D μ , u , h Φ ^ ( μ ¯ , u ¯ , i Ω ) acts on

( μ ˙ , u ˙ , h ˙ ) ∈ R × C 0 2 , r ( Ω ¯ ) × C 4 ( Ω ¯ ; R N )

as follows (see Theorem 2.2 in [11]):

(3.1) D μ , u , h Φ ^ ( μ ¯ , u ¯ , i Ω ) [ μ ˙ , u ˙ , h ˙ ] = Δ u ˙ + μ ¯ f ¯ ′ u ˙ + f ¯ μ ˙ + h ˙ ⋅ ∇ ( Δ u ¯ + μ ¯ f ¯ ) − ( Δ + μ ¯ f ¯ ′ ) h ˙ ⋅ ∇ u ¯ = ( Δ + μ ¯ f ¯ ′ ) u ˙ − ( Δ + μ ¯ f ¯ ′ ) h ˙ ⋅ ∇ u ¯ + f ¯ μ ˙ ,

where we used the fact that Δ u ¯ + μ ¯ f ¯ = Φ ^ ( μ ¯ , u ¯ , i Ω ) = 0 .

At this point, observe that, by the Fredholm property of the operator Δ + μ ¯ f ¯ on C 0 2 , r ( Ω ¯ ) , we have that the subspace { D μ , u , h Φ ^ ( μ ¯ , u ¯ , i Ω ) [ ( 0 , u ˙ , 0 ) ] , u ˙ ∈ C 0 2 , r ( Ω ¯ ) } , is closed and has finite codimension. Next, since u ¯ ∈ C 0 2 , r ( Ω ¯ ) and ∂ Ω is of class C 4 , then by standard elliptic regularity theory, we find that u ¯ ∈ C 0 3 , r ( Ω ¯ ) and then h ˙ ⋅ ∇ u ¯ ∈ C 2 , r ( Ω ¯ ) . As a consequence, we can prove that the subspace { D μ , u , h Φ ^ ( μ ¯ , u ¯ , i Ω ) [ ( 0 , 0 , h ˙ ) ] , h ˙ ∈ C 4 ( Ω ¯ ; R N ) } is closed with finite codimension as well. Indeed, let us define K : C 2 , r ( Ω ¯ ) ↦ C 2 , r ( Ω ¯ ) as the linear operator, which, to any ϕ ∈ C 2 , r ( Ω ¯ ) , associates the unique solution ϕ b = K [ ϕ ] of Δ ϕ b = 0 , ϕ b = ϕ on ∂ Ω . Clearly, this is always well posed since Ω is of class C 4 and ϕ ∈ C 2 , r ( Ω ¯ ) . Then,

Δ ϕ + μ ¯ f ¯ ′ ϕ = g ∈ C r ( Ω ¯ ) ,

if and only if

ϕ ∈ C 2 , r ( Ω ¯ ) and ϕ + T [ ϕ ] = G [ g ] ∈ C 2 , r ( Ω ¯ ) ,

where G [ g ] = ∫ Ω G ( x , y ) g ( y ) and T : C 2 , r ( Ω ¯ ) ↦ C 2 , r ( Ω ¯ ) , T ( ϕ ) = G [ μ ¯ f ¯ ′ ϕ ] − K [ ϕ ] . Since Ω is of class C 4 , then by standard elliptic estimates, T maps C 2 , r ( Ω ¯ ) into C 3 , r ( Ω ¯ ) . Therefore, T is compact, and then, we conclude by the Fredholm alternative that the range of ( Δ + μ ¯ f ¯ ′ ) ( h ˙ ⋅ ∇ u ¯ ) , h ˙ ⋅ ∇ u ¯ ∈ C 2 , r ( Ω ¯ ) , is closed in C r ( Ω ¯ ) and has finite codimension.

At this point, we deduce from these two facts that there exists a nontrivial ϕ ⊥ ∈ C r ( Ω ¯ ) , which is orthogonal to the image of D μ , u , h Φ ^ ( μ ¯ , u ¯ , i Ω ) , that is,

(3.2) ∫ Ω ϕ ⊥ ( ( Δ + μ ¯ f ¯ ′ ) u ˙ − ( Δ + μ ¯ f ¯ ′ ) h ˙ ⋅ ∇ u ¯ + f ¯ μ ˙ ) = 0 , ∀ ( μ ˙ , u ˙ , h ˙ ) .

Putting ( μ ˙ , u ˙ , h ˙ ) = ( μ ˙ , 0 , 0 ) in (3.2), we find ∫ Ω f ¯ ϕ ⊥ = 0 , and then if we choose h ˙ = 0 , we find that

∫ Ω ϕ ⊥ ( Δ + μ ¯ f ¯ ′ ) u ˙ = 0 , ∀ u ˙ ∈ C 0 2 , r ( Ω ¯ ) ,

which shows that ϕ ⊥ is a C r ( Ω ¯ ) distributional solution of Δ ϕ ⊥ + μ ¯ f ¯ ′ ϕ ⊥ = 0 . Therefore, by standard elliptic estimates (where we recall that ∂ Ω is of class C 4 ), ϕ ⊥ is a C 0 2 ( Ω ¯ ) solution of Δ ϕ ⊥ + μ ¯ f ¯ ′ ϕ ⊥ = 0 . As a consequence we observe that (3.2) is reduced to

∫ Ω ϕ ⊥ ( Δ + μ ¯ f ¯ ′ ) h ˙ ⋅ ∇ u ¯ = 0 , ∀ h ˙ ∈ C 4 ( Ω ¯ ; R N ) ,

which allows us to deduce that

0 = ∫ Ω ϕ ⊥ ( Δ + μ ¯ f ¯ ′ ) h ˙ ⋅ ∇ u ¯ = ∫ Ω ϕ ⊥ ( Δ + μ ¯ f ¯ ′ ) h ˙ ⋅ ∇ u ¯ − ∫ Ω ( Δ ϕ ⊥ + μ ¯ f ¯ ′ ϕ ⊥ ) h ˙ ⋅ ∇ u ¯ = ∫ Ω ϕ ⊥ Δ ( h ˙ ⋅ ∇ u ¯ ) − ∫ Ω ( Δ ϕ ⊥ ) h ˙ ⋅ ∇ u ¯ = ∫ ∂ Ω ( ϕ ⊥ ∂ ν ( h ˙ ⋅ ∇ u ¯ ) − h ˙ ⋅ ∇ u ¯ ( ∂ ν ϕ ⊥ ) ) = − ∫ ∂ Ω ( ∂ ν ϕ ⊥ ) h ˙ ⋅ ∇ u ¯ = − ∫ ∂ Ω ( ∂ ν ϕ ⊥ ) ( ∂ ν u ¯ ) h ˙ ⋅ ν , ∀ h ˙ ∈ C 4 ( Ω ¯ , R N ) .

Therefore, since h ˙ is arbitrary, we conclude that,

( ∂ ν ϕ ⊥ ) ( ∂ ν u ¯ ) ≡ 0 on ∂ Ω .

At this point, we observe that since f ¯ > 0 on Ω ¯ and u ¯ = 0 on ∂ Ω , then, by the strong maximum principle, we have u ¯ > 0 in Ω . Since ∂ Ω is of class C 4 , we can apply the Hopf boundary lemma and conclude that ∂ ν u ¯ < 0 on ∂ Ω . Therefore, we conclude that necessarily ∂ ν ϕ ⊥ ≡ 0 on ∂ Ω , which is in contradiction with the Hopf boundary lemma. This contradiction shows that ( i i ) holds, and then, we can apply Theorem 3.4 and conclude that there exists a meager set ℱ ⊂ Diff 4 ( Ω 0 ) such that if h ( Ω 0 ) ∉ ℱ , then η = 0 is a regular value of Φ ( μ , v , h ) .

Step 2: We have from step 1 that there exists a meager set ℱ ⊂ Diff 4 ( Ω 0 ) such that if h ∉ ℱ and Ω ≔ h ( Ω 0 ) , then η = 0 is a regular value of the map Φ ( ⋅ , ⋅ , h ) . As a consequence, for any ( μ ¯ , v ¯ ) , which solves

Φ ( μ , v ) = F Ω ( μ , v ) = 0

and setting f ¯ = f ( v ¯ ) , f ¯ ′ ( v ¯ ) , then the differential

L ¯ [ μ ˙ , v ˙ ] ≔ D μ , v Φ ( μ ¯ , v ¯ ) [ μ ˙ , v ˙ ] = Δ v ˙ + μ ¯ f ¯ ′ ( v ¯ ) v ˙ + f ¯ μ ˙ .

is surjective. On the other side, since v ¯ solves (1.1), then the operator,

Δ v ˙ + μ ¯ f ¯ ′ ( v ¯ ) v ˙

is just L μ ¯ for which the Fredholm alternative holds. Let us define R = R ( L μ ¯ ) ⊆ C r ( Ω ¯ ) to be the range of L μ ¯ . Now if L μ ¯ is surjective, then, by the Fredholm alternative, we have Ker ( L μ ¯ ) = ∅ , which is ( a ) in the statement of Theorem 1.1. Therefore, we can assume without loss of generality that L μ ¯ is not surjective, let d ¯ = codim ( R ) be the codimension of R . Since L ¯ is surjective, by the Fredholm alternative, it is not difficult to see that d ¯ ≤ 1 , and since L μ ¯ is not surjective, then necessarily d ¯ = 1 . We will conclude the proof by showing that ( b ) holds in this case. Indeed, obviously, the kernel must be one-dimensional, Ker ( L ¯ ) = span { ϕ ¯ } , for some ϕ ¯ ∈ C 0 2 , r ( Ω ¯ ) , which satisfies L μ ¯ [ ϕ ¯ ] = 0 . Since L ¯ is surjective, then f ¯ ′ ϕ ¯ must be an element of its range and then there exists ϕ ∈ C 0 2 , r ( Ω ¯ ) , which satisfies

L μ ¯ [ ϕ ] + μ ˙ f ¯ = f ¯ ′ ϕ ¯ .

Multiplying this equation by ϕ ¯ and integrating by parts, we find that

μ ˙ ∫ Ω f ¯ ϕ ¯ = ∫ Ω f ¯ ′ ϕ ¯ 2 ,

and since f ′ ( t ) > 0 , ∀ t , then we deduce that necessarily ∫ Ω f ¯ ϕ ¯ ≠ 0 . In other words, ( b ) of Theorem 1.1 holds and the proof is concluded.□

We are ready to present the proof of Theorem 1.2.

Proof of Theorem 1.2

It is well-known [4] that, due to (H1), there exists μ ∗ < + ∞ such that μ ≤ μ ∗ for any solution of (1.1) and in particular that there exists a continuous simple curve of solutions of (1.1) (the branch of minimal solutions) for any μ < μ ∗ which emanates from ( μ , v μ ) = ( 0 , 0 ) , which we denote by G ( Ω ) . In particular, with the notations of Section 2, G ( Ω ) is characterized by the fact that the first eigenvalue of the linearized equation, which we denote by σ 1 ( μ , v μ ) , satisfies σ 1 ( μ , v μ ) > 0 for any ( μ , v μ ) ∈ G ( Ω ) . In view of ( H2 ) and standard elliptic theory, we have that v ∗ = v μ ∣ μ = μ ∗ is a classical solution and σ 1 ( μ ∗ , v ∗ ) = 0 . By Theorem 1.1, we have that ( b ) holds for ( μ ∗ , v ∗ ) , and then by Proposition 2.2, we can continue G ( Ω ) to a continuous and simple curve without bifurcation points, [ 0 , s 1 + δ 1 ) ∋ s ↦ ( μ ( s ) , v ( s ) ) , which locally around any point s 0 > 0 admits a real analytic reparametrization, that is, an injective and continuous map γ 0 : ( − 1 , 1 ) → ( s 0 − ε , s 0 + ε ) , s = γ 0 ( t ) , such that γ 0 ( 0 ) = s 0 and ( μ ( γ 0 ( t ) ) , v ( γ 0 ( t ) ) ) is real analytic. Therefore, locally, this branch has also the structure of a one-dimensional real analytic manifold and we denote it by,

G ( s 1 + δ 1 ) = { [ 0 , s 1 + δ 1 ) ∋ s ↦ ( μ ( s ) , v ( s ) ) } ,

which satisfies

G ( s 1 + δ 1 ) ¯ = { [ 0 , s 1 + δ 1 ] ∋ s ↦ ( μ ( s ) , v ( s ) ) } ,

where, for some s 1 > 0 and δ 1 > 0 , we have:

( μ ( s ) , v ( s ) ) is continuous and locally (up to reparametrization) real analytic for s ∈ [ 0 , s 1 + δ 1 ] ;
v ( s ) is a solution of (1.1) with μ = μ ( s ) for any s ∈ [ 0 , s 1 + δ 1 ] ;
μ ( s ) = s for s ≤ s 1 , μ ( s 1 ) = μ ∗ ;
the inclusion { ( μ , v μ ) , μ ∈ [ 0 , μ ∗ ] } ≡ G ( Ω ) ¯ ⊂ G ( s 1 + δ 1 ) , holds;
inf [ s 1 , s 1 + δ 1 ) μ ( s ) > 0 and 0 < μ ( s ) ≤ μ ∗ , ∀ s ∈ ( 0 , s 1 + δ 1 ) ;
0 ∉ Σ ( L μ ( s ) ) , ∀ s ∈ ( 0 , s 1 + δ 1 ) ⧹ { s 1 } ,
Ker ( L μ ( s 1 ) ) = span { ϕ 1 } and ∫ Ω f ( v ( s 1 ) ) ϕ 1 ≠ 0 ,

where Σ ( L μ ( s ) ) denotes the spectrum of L μ ( s ) . Clearly, ( A 1 ) 6 follows from ( b ) of Theorem 1.1. Concerning ( A 1 ) 5 , we recall that, by Proposition 2.2, either σ 1 ( s ) vanishes identically around s 1 or its zero must be isolated. In particular, since σ 1 ( s ) is (locally up to a reparametrization) real analytic, its level sets cannot have accumulation points unless σ 1 ( s ) is locally constant and consequently unless it is constant on [ 0 , s 1 + δ 1 ) . However, we can rule out this case since, in view of ( A 1 ) 2 , for s < s 1 , we have σ 1 ( μ ( s ) , v ( s ) ) > 0 and then no σ k ( s ) can vanish identically, which shows that ( A 1 ) 5 holds as well. Therefore, it is well defined

s 2 ≔ sup { t > s 1 : inf s ∈ [ s 1 , t ) μ ( s ) > 0 , 0 ∉ Σ ( L μ ( s ) ) , ∀ ( μ ( s ) , v ( s ) ) ∈ G ( t ) , ∀ s 1 < s < t } .

At this point, either inf s ∈ [ s 1 , s 2 ) μ ( s ) = 0 or inf s ∈ [ s 1 , s 2 ) μ ( s ) > 0 .

If inf s ∈ [ s 1 , s 2 ) μ ( s ) = 0 , we set s ∞ = s 2 ,

(3.3) G ( s ∞ ) = { [ 0 , s ∞ ) ∋ s ↦ ( μ ( s ) , v ( s ) ) } ,

and claim that in this case, necessarily μ ( s ) → 0 + and ‖ v ( s ) ‖ ∞ → + ∞ as s → s ∞ .

We first prove that μ ( s ) → 0 + and argue by contradiction, assuming that there exists a sequence { s n } ⊂ ( 0 , s ∞ ) such that s n → s ∞ , as n → + ∞ and μ ( s n ) ≥ δ > 0 for some δ > 0 . In view of ( A 1 ) 4 , passing to a subsequence if necessary, we can assume that μ ( s n ) → μ ¯ ∈ [ δ , μ ∗ ] . By ( H2 ) and passing to a further subsequence, we would deduce that v ( s n ) → v ¯ , where ( μ ¯ , v ¯ ) is a solution of (1.1). By Theorem 1.1, we see that either ( a ) or ( b ) holds and then, possibly with the aid of Proposition 2.2, we would deduce that locally around ( μ ¯ , v ¯ ) , the set of solutions of (1.1) is a real analytic parametrization of the form ( μ ¯ ( t ) , v ¯ ( t ) ) , t ∈ ( − ε , ε ) for some ε > 0 with ( v ¯ ( 0 ) , v ¯ ( 0 ) ) = ( μ ¯ , v ¯ ) . In particular, for any fixed n ¯ large enough, we can assume without loss of generality that ( μ ¯ ( t ) , v ¯ ( t ) ) , t ∈ ( − ε , 0 ) coincides with ( μ ( s ) , v ( s ) ) , s ∈ ( s n ¯ , s ∞ ) . Now by construction, μ ( s ) > 0 in [ s 1 , s n ¯ ] , and since μ ( s ) is continuous, we have inf s ∈ [ s 1 , s n ¯ ] μ ( s ) ≥ δ ¯ > 0 for some δ ¯ > 0 . On the other side, possibly taking a larger s n ¯ , we have inf s ∈ [ s n ¯ , s ∞ ) μ ( s ) ≥ δ 2 . In other words, we have a contradiction to inf s ∈ [ s 1 , s ∞ ) μ ( s ) = 0 and the claim is proved.

Next, we show that ‖ v ( s ) ‖ ∞ → + ∞ and argue by contradiction. If this was not the case, we could find a sequence { s n } ⊂ ( 0 , s ∞ ) such that s n → s ∞ , as n → + ∞ and ‖ v ( s n ) ‖ ∞ ≤ C for some C > 0 . Since we have shown that μ ( s ) → 0 + as s → s ∞ , then passing to a subsequence, we would deduce that v ( s n k ) → v ¯ , where v ¯ solves (1.1) with μ = 0 . However by ( A 1 ) 4 , this fact implies that ( μ , v μ ) = ( 0 , 0 ) would be a bifurcation point, which is clearly impossible, which proves the claim. At this point, since by definition, ℛ ∞ is a closed and connected set, it is not difficult to see that ℛ ∞ ≡ G ( s ∞ ) .

After a suitable reparametrization, we can assume without loss of generality that s 2 = + ∞ and we conclude that statement of Theorem 1.2 is true as far as inf s ∈ ( 0 , s 2 ) μ ( s ) = 0 . Therefore, we can assume without loss of generality that inf s ∈ ( 0 , s 2 ) μ ( s ) > 0 . In this case, in view of ( A 1 ) 4 , (H2), Theorem 1.1, and Proposition 2.2, it is not difficult to see that ( μ ( s ) , v ( s ) ) converges to a solution ( μ 2 , v 2 ) as s → s 2 and that 0 ∈ Σ ( L μ 2 ) , and in particular that we can continue the branch G ( s 2 ) in a right neighborhood of s 2 to a continuous curve, which admits local real analytic reparametrizations. In particular, by arguing as above, we see that 0 ∉ Σ ( L μ ( s ) ) for s ∉ { s 1 , s 2 } and we can argue by induction defining, for k ≥ 3 ,

s k ≔ sup { t > s k − 1 : inf s ∈ [ s 1 , t ) μ ( s ) > 0 , 0 ∉ Σ ( L μ ( s ) ) , ∀ ( μ ( s ) , v ( s ) ) ∈ G ( t ) , ∀ s k − 1 < s < t } .

If there exists some k ≥ 3 such that inf s ∈ ( 0 , s k ) μ ( s ) = 0 , then as for (3.3) we are done. Otherwise by using ( A 1 ) 4 , (H2), Theorem 1.1, and Proposition 2.2, we can find sequences s k and δ k > 0 such that, for any k ∈ N we have, s k + 1 > s k > ⋯ > s 2 > s 1 , s k + δ k < s k + 1 and

( A k ) 0 ( μ ( s ) , v ( s ) ) is continuous and simple curve without bifurcation points (which admits local real analytic reparametrizations) defined for s ∈ [ 0 , s k + δ k ] ;

( A k ) 1 v ( s ) is a solution of (1.1) with μ = μ ( s ) for any s ∈ [ 0 , s k + δ k ] ;

( A k ) 2 μ ( s ) = s for s ≤ s 1 , μ ( s 1 ) = μ ∗ ;

( A k ) 3 the inclusion { ( μ ( s ) , v ( s ) ) , s ∈ [ 0 , s k ] } ≡ G ( s k ) ( Ω ) ¯ ⊂ G ( s k + δ k ) , holds;

( A k ) 4 inf [ s 1 , s k + δ k ) μ ( s ) > 0 and 0 < μ ( s ) ≤ μ ∗ , ∀ s ∈ ( 0 , s k + δ k ) ;

( A k ) 5 0 ∉ Σ ( L μ ( s ) ) , ∀ s ∈ ( 0 , s k + δ k ) ⧹ { s 1 , s 2 , … , s k } ;

( A k ) 6 Ker ( L λ ( s k ) ) = span { ϕ k } and ∫ Ω f ( v ( s k ) ) ϕ k ≠ 0 .

Let s ∞ = lim k → + ∞ s k , we claim that:

Claim: μ ( s ) → 0 + as s → s ∞ .

We argue by contradiction and assume that along an increasing sequence { s ^ j } such that s ^ j → s ∞ , it holds μ ( s ^ j ) ≥ δ > 0 for some δ > 0 . Clearly, we can extract a subsequence { s k j } ⊂ { s k } such that s k j < s ^ j ≤ s k j + 1 . By ( A k ) 4 and ( H2 ) , we can extract an increasing subsequence (which we will not relabel) such that ( μ ( s ^ j ) , v ( s ^ j ) ) converges to a solution ( μ ^ , v ^ ) of (1.1) as j → + ∞ , where δ ≤ μ ^ ≤ μ ∗ .

By Theorem 1.1, we can apply either Lemma 2.1 or Proposition 2.2 and conclude that locally around ( μ ^ , v ^ ) the set of solutions of (1.1) is a real analytic parametrization of the form ( μ ^ ( t ) , v ^ ( t ) ) , t ∈ ( − ε , ε ) for some ε > 0 with ( μ ^ ( 0 ) , v ^ ( 0 ) ) = ( μ ^ , v ^ ) . In particular for j large enough, we can assume without loss of generality that ( μ ^ ( t ) , v ^ ( t ) ) , t ∈ ( − ε , 0 ) coincides with ( μ ( s ) , v ( s ) ) , s ∈ ( s j ^ , s ∞ ) . Let { σ ^ n } n ∈ N be the eigenvalues corresponding to ( μ ^ , v ^ ) and { σ ^ n ( t ) } n ∈ N be those corresponding to ( μ ^ ( t ) , v ^ ( t ) ) . On the one side, since by construction 0 ∈ σ ( L λ ( s k j ) ) and s k j < s ^ j ≤ s k j + 1 for any j , then we have that 0 ∈ σ ( L λ ^ ) . Indeed, if this was not the case, then, by Lemma 2.1 and since the eigenvalues are isolated, we would have that there exists a fixed full neighborhood of 0 with empty intersection with σ ( L λ ( s k j ) ) for any j large enough, which is a contradiction since the number of negative eigenvalues is, locally around each positive solution, uniformly bounded. As a consequence, there exists n ∈ N such that σ ^ n = 0 . On the other side, since σ ^ n ( t ) is, in particular, a continuous function of t , by using once more the fact that the eigenvalues are isolated, possibly passing to a further subsequence if necessary, we must obviously have σ ^ n ( t j ^ ) = 0 for some t j ^ → 0 − as j → + ∞ . Whence σ ^ n must vanish identically in ( − ε , 0 ] . In particular, the n th eigenvalue relative to ( μ ( s ) , v ( s ) ) must vanish identically for s ∈ ( s j ^ , s ∞ ) and therefore in [ 0 , s ∞ ) . This is again a contradiction to ( A k ) 2 since for s < s 1 we have σ 1 ( μ ( s ) , v ( s ) ) > 0 and then no eigenvalue can vanish identically. Therefore, a contradiction arises, which shows that μ ( s ) → 0 + as s → s ∞ .

At this point, arguing as above, it is not difficult to see that ‖ v ( s ) ‖ ∞ → + ∞ as s → s ∞ and, defining

G ( s ∞ ) = { [ 0 , s ∞ ) ∋ s ↦ ( μ ( s ) , v ( s ) ) } ,

that ℛ ∞ ≡ G ( s ∞ ) . After a suitable reparametrization, we can assume without loss of generality that s 2 = + ∞ , which concludes the proof.□

Acknowledgments

Daniele Bartolucci would like to express his warmest thanks to N. Dancer for pointing out that a combined use of Lemmas 9 and 10 in [6] shows that ℛ ∞ is path-connected and to B. Buffoni for very useful discussions about analytic global bifurcation theory.

Funding information: D. Bartolucci is partially supported by Beyond Borders project 2019 (sponsored by University of Rome “Tor Vergata”) “Variational Approaches to PDEs,” MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. W. Yang is partially supported by NSFC No. 12171456 and No. 11871470.
Conflict of interest: The authors state no conflict of interest.

References

[1] D. Bartolucci and A. Jevnikar, On the global bifurcation diagram of the Gelafand problem, Analysis and P.D.E. 14-8 (2021), 2409–2426. 10.2140/apde.2021.14.2409Search in Google Scholar

[2] B. Buffoni, E. N. Dancer, and J. F. Toland, The sub-harmonic bifurcation of Stokes waves, Arch. Rat. Mech. Anal. 152 (2000), no. 3, 24–271. 10.1007/s002050000087Search in Google Scholar

[3] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton University Press, Princeton, 2003. 10.1515/9781400884339Search in Google Scholar

[4] M. G. Crandall and P. H. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. An. 58 (1975), 207–218. 10.1007/BF00280741Search in Google Scholar

[5] N. Dancer, Real analyticity and non-degeneracy, Math. Ann. 325 (2003), 369–392. 10.1007/s00208-002-0352-2Search in Google Scholar

[6] N. Dancer, Global structure of the solutions of non-linear real analytic eigenvalue problems, Proc. Lond. Math. Soc. 27 (1973), no. 3, 747–765. 10.1112/plms/s3-27.4.747Search in Google Scholar

[7] N. Dancer, On the structure of solutions of an equation in catalysis theory when a parameter is large, J. Diff. Equ. 37 (1980), no. 3, 404–437. 10.1016/0022-0396(80)90107-2Search in Google Scholar

[8] D. G. de Figuereido, P.-L. Lions, and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), no. 1, 41–63. Search in Google Scholar

[9] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal. 49 (1972/73), 241–269. 10.1007/BF00250508Search in Google Scholar

[10] M. Holzmann and H. Kielhooofer, Uniqueness of global positive solution branches of nonlinear elliptic problems, Math. Ann. 300 (1994), 221–241. 10.1007/BF01450485Search in Google Scholar

[11] D. Henry, Perturbation of the boundary in boundary-value problems of partial differential equations, London Math. Soc. L.N.S., vol. 318, Cambridge University Press, Cambridge, 2005. 10.1017/CBO9780511546730Search in Google Scholar

[12] D. Henry, Generic properties of equilibrium solutions by perturbation of the boundary. In: S-N Chow and J. K. Hale(Ed.), Dynamics of Infinite Dimensional Systems, Springer, Berlin, Heidelberg, 1987. 10.1007/978-3-642-86458-2_15Search in Google Scholar

[13] T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, Heidelberg, 1966. 10.1007/978-3-662-12678-3Search in Google Scholar

[14] P. Korman, Global Solution Curves for Semilinear Elliptic Equations, World Scientific, Singapore, 2012. 10.1142/8308Search in Google Scholar

[15] P. Korman, Global solution curves for self-similar equations, J. Diff. Equ. 257 (2014), 2543–2564. 10.1016/j.jde.2014.05.045Search in Google Scholar

[16] P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, S.I.A.M. Review 24 (1982), no. 4, 441–467. 10.1137/1024101Search in Google Scholar

[17] A. M. Micheletti, Perturbazione dello spetro di un operatore ellitico di tipo variazionale, in relazione ad una variazione del campo, Annali Mat. Pura App. 97 (1973), 267–281. 10.1007/BF02414915Search in Google Scholar

[18] S. Nakane A bifurcation phenomenon for a Dirichlet problem with an exponential nonlinearity, J. Math. An. App. 161 (1991), 227–240. 10.1016/0022-247X(91)90372-7Search in Google Scholar

[19] K. Nagasaki and T. Suzuki, Radial and nonradial solutions for the nonlinear eigenvalue problem Δu+λeu=0 on annuli in R2, J. Differential Equations 87 (1990), 144–168. 10.1016/0022-0396(90)90020-PSearch in Google Scholar

[20] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigen-values problems with exponentially dominated nonlinearities, Asymptotic Analysis 3 (1990), 173–188. 10.3233/ASY-1990-3205Search in Google Scholar

[21] M Plum and C. Wieners, New solutions of the Gelfand problem, J. Math. An. App. 269 (2002), 588–606. 10.1016/S0022-247X(02)00038-0Search in Google Scholar

[22] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1981), 487–513. 10.1016/0022-1236(71)90030-9Search in Google Scholar

[23] B. P. Rynne, The structure of Rabinowitz’ global bifurcating continua for generic quasilinear elliptic equations, Noniln. An. T.M.A. 32 (1998), 167–181. 10.1016/S0362-546X(97)00471-9Search in Google Scholar

[24] J. C. Saut and R. Temam, Generic properties of nonlinear boundary-value problems, Comm. P.D.E. 4 (1979), 293–319. 10.1080/03605307908820096Search in Google Scholar

[25] S. Smale, An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87 (1969), 861–866. 10.2307/2373250Search in Google Scholar

[26] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), 1059–1078. 10.2307/2374041Search in Google Scholar

Received: 2022-09-04

Revised: 2023-03-17

Accepted: 2023-03-17

Published Online: 2023-04-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2022-0062

Keywords for this article

Rabinowitz continuum; bifurcation analysis; generic properties

Creative Commons

BY 4.0