Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 21–31 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Existence of many positive nonradial solutions for a superlinear Dirichlet problem on thin annuli ^{∗}

Alfonso Castro & Marcel B. Finan

Abstract

We study the existence of many positive nonradial solutions of a superlinear Dirichlet problem in an annulus in ℝ^{N}. Our strategy consists of finding the minimizer of the energy functional restricted to the Nehrai manifold of a subspace of functions with symmetries. The minimizer is a global critical point and therefore is a desired solution. Then we show that the minimal energy solutions in different symmetric classes have mutually different energies. The same approach has been used to prove the existence of many sign-changing nonradial solutions (see [5]).

1 Introduction

In this article we study the existence of many positive nonradial solutions of the equation

Δu + f(u) = 0 in Ω_{ϵ}
u > 0 in Ω_{ϵ}  (1)
u = 0 on ∂Ω_{ϵ} ,

where

Ω_{ϵ} = {x ∈ ℝ^{3} : 1 − ϵ < |x| < 1}.

The non-linearity f is of class C^{1}(ℝ) and satisfies the following conditions:

(C1) f(0) = 0 and f′(0) < λ_{1}, where λ_{1} is the smallest eigenvalue of −Δ with zero Dirichlet boundary condition in Ω_{ϵ}.

(C2) f′(u) > \frac{f(u)}{u} for all u ≠ 0.

(C3) (Superlinearity) lim_{|u|→∞} \frac{f(u)}{u} = ∞.

∗Mathematics Subject Classifications: 35J20, 35J25, 35J60 Key words: Superlinear Dirichlet problem, positive nonradial solutions, variational methods \begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University.

Published October 24, 2000.

(C4) (Subcritical growth) There exist constants C > 0 and p ∈ (1, 5) such

that

|f′(u)| ≤ C(|u|^{p−1} + 1), ∀u ∈ ℝ.

(C5) There exist constants m ∈ (0, 1) and ρ such that for |u| > ρ,

uf(u) ≥ \frac{2}{m}F (u) > 0,

where F (u) = ∫_{0}^{u} f(s) ds.

This paper is motivated by the work of Coffman [6], Li [7] and Lin [8]. In 1984, Coffman showed that for f(t) = −t + t^{p}, where p = 2N + 1, N = 2, the above problem has many positive nonradial solutions. His result was extended by Li [7] to the N-dimensional case with N ≥ 4 or N = 2 and for p ∈ (1, \frac{N+2}{N−2}), when the nonlinear term is λt + t^{p} for λ ≤ 0. Our main result (see Theorem 1.1 below) concerns the case N = 3. We will show that for N = 3 our problem has many distinct nonradial solutions. We follow the strategy used in [6, 7] and [8]. That is, we look for the minimizer of the energy functional restricted to the Nehari manifold of a subspace of functions with symmetries. The minimizer is a global critical point and therefore is a solution to (1). Then we show that the minimal energy solutions in different symmetric classes has mutually different energies.

We would like to point out here that during the preparation of this article we were unaware of the papers by Byeon [1] and Catrina and Wang [2], where the above problem has been solved. However, our approach is different from theirs.

Our main result is the following

Theorem 1.1 Let conditions (C1) - (C5) be satisfied. Then, for each positive integer k there exists ϵ_{1}(k) > 0 such that if 0 < ϵ < ϵ_{1}(k) then (1) has k distinct positive nonradial solutions.

We note that our argument works for N ≥ 2 and not only for the three dimensional case.

The approach used in proving Theorem 1.1 is similar to the one used in [7], i.e., we consider the functionals

J(u) = ∫_{Ω_{ϵ}} \{\frac{1}{2}|∇u|^{2} − F (u)\}dx

and

γ(u) = ∫_{Ω_{ϵ}} (|∇u|^{2} − uf(u)) dx

on H_{0}^{1}(Ω_{ϵ}), and for k ≥ 1, we consider the space of invariant functions

H(ϵ, k) := Fix(G_{k} × {id_{ℝ}, −id_{ℝ}})
= {v ∈ H_{0}^{1}(Ω_{ϵ}) : v(g(x_{1}, x_{2}), T x_{3}) = v(x_{1}, x_{2}, x_{3}),

∀ (g, T) ∈ G_{k} × {id_{ℝ}, −id_{ℝ}}}
= {v ∈ H_{0}^{1}(Ω_{ϵ}) : v(x_{1}, x_{2}, x_{3}) = u(x_{1}, x_{2}, |x_{3}|) for some u
satisfying u(g(x_{1}, x_{2}), |x_{3}|) = u(x_{1}, x_{2}, |x_{3}|) ∀g ∈ G_{k}}

where G_{k} = {g^{i} : 0 ≤ i ≤ k − 1} and gz = e^{2πi/k}z, z ∈ ℂ ≃ ℝ^{2}. Also, we consider the Nehari manifold

S(ϵ, k) = {v ∈ H(ϵ, k)\{0} : γ(v) = 0}.

Similarly, we define the space of functions

H(ϵ, ∞) := O(2) × {id_{ℝ}, −id_{ℝ}}
= {v ∈ H_{0}^{1}(Ω_{ϵ}) : v(g(x_{1}, x_{2}), T x_{3}) = v(x_{1}, x_{2}, x_{3}),
∀(g, T) ∈ O(2) × {id_{ℝ}, −id_{ℝ}}}
= \{v ∈ H_{0}^{1}(Ω_{ϵ}) : v(x_{1}, x_{2}, x_{3}) = u(\sqrt{x_{1}^{2} + x_{2}^{2}}, |x_{3}|), for some u\},

and the manifold

S(ϵ, ∞) = {v ∈ H(ϵ, ∞)\{0} : γ(v) = 0},

where O(2) denotes the space of 2 × 2 orthogonal matrices. Note that if u ∈ H(ϵ, ∞) then u is θ−independent.

Associated with the above sets, we consider the numbers

j_{k}^{ϵ} = inf_{v∈S(ϵ,k)} J(v) and j_{∞}^{ϵ} = inf_{v∈S(ϵ,∞)} J(v).

We prove Theorem 1.1 by establishing the following lemmas:

Lemma 1.2 For k = 1, 2, ..., ∞, j_{k}^{ϵ} is achieved by some u_{ϵ,k} ∈ S(ϵ, k) and u_{ϵ,k} is a critical point of J on H(ϵ, k).

Lemma 1.3 Let u_{ϵ,k} be as in Lemma 1.2. Then u_{ϵ,k} is a critical point of J on H_{0}^{1}(Ω_{ϵ}).

Lemma 1.4 Given a positive integer k, there exists ϵ_{1}(k) such that for 0 < ϵ < ϵ_{1}(k) we have

j_{k}^{ϵ} < j_{∞}^{ϵ}.

Lemma 1.5 For n = 2, 3, 4, ..., k = 1, 2, 3, ..., j_{kn}^{ϵ} < j_{∞}^{ϵ} implies j_{k}^{ϵ} < j_{kn}^{ϵ}.

This article is organized as follows: In Section 2, we prove Lemmas 1.2–1.5. In Section 3, we prove Theorem 1.1.

2 Proof of Lemmas 1.2–1.5

Proof of Lemma 1.2: The proof follows from [3] combined with the facts that the embeddings H(ϵ, k) ⊂ L^{p}(Ω_{ϵ}) and H(ϵ, ∞) ⊂ L^{∞}(Ω_{ϵ}) are compact (See [9]) ♢

Lemma 1.3 is a result of the symmetric criticality principle: if u_{ϵ,k} is a critical point of J on H(ϵ, k), then u_{ϵ,k} is a critical point of J on H_{0}^{1}(Ω_{ϵ}) (See [9]).

To prove Lemma 1.4 we need the following results.

Lemma 2.1 (Poincaré’s inequality) Let Ω ⊂ ℝ^{N} be a smooth bounded domain with diameter d. Then for u ∈ H_{0}^{1}(Ω) we have

∫_{Ω} u^{2}dx ≤ \frac{d^{2}}{N} ∫_{Ω} |∇u|^{2}dx .

Lemma 2.2 ([3]) 0 is a local minimum of J. If u ∈ H(ϵ, k) − {0}, then there exists a unique α = α(u) ∈ (0, ∞) such that αu ∈ S(ϵ, k). Moreover, J(αu) = max_{λ>0} J(λu) > 0.

Lemma 2.3 ([4]) For |v| > ρ and s > 2 we have

svf(sv) ≥ Cs^{2/m}vf(v)

for some constant C > 0.

Proof of Lemma 1.4: Let k ≥ 1 be an integer and ϵ > 0 to be chosen below. According to Lemma 1.2, there exist u_{ϵ,k} ∈ S(ϵ, k) and u_{ϵ,∞} ∈ S(ϵ, ∞) such that j_{k}^{ϵ} = J(u_{ϵ,k}) and j_{∞}^{ϵ} = J(u_{ϵ,∞}). By Lemma 1.3, u_{ϵ,k} and u_{ϵ,∞} are solutions to Problem 1. Let Ω_{ϵ}^{k} be the set of points x = (x_{1}, x_{2}, x_{3}) ≡ (r, θ, x_{3}) ∈ Ω_{ϵ} such that θ ∈ [0, 2π/k].

Let j be a positive integer to be chosen independent of k and ϵ. Define

ω(r, θ, x_{3}) = \{ \begin{matrix}u_{ϵ,∞}(r, |x_{3}|) sin (jkθ) & for θ ∈ [0, π/(jk)] \\ 0 & for θ ∈ [π/(jk), 2π/k] , \end{matrix}

where r = (x_{1}^{2} + x_{2}^{2})^{1/2}. Extend ω periodically to all of Ω_{ϵ} in the θ direction. Let z ∈ H(ϵ, k) be the resulting extension. Since u_{ϵ,∞} > 0 then z > 0. By the chain rule we have

|∇ω(r, θ, x_{3})|^{2} = (u_{ϵ,∞})_{r}^{2} sin^{2} (jkθ) + \frac{1}{r^{2}} (u_{ϵ,∞})^{2}(jk)^{2} cos^{2} (jkθ)
+|∇_{x_{3}}u_{ϵ,∞}|^{2} sin^{2} (jkθ)

if θ ∈ [0, π/(jk)], otherwise ∇ω = 0. Thus,

|∇ω(r, θ, x_{3})|^{2} ≤ (u_{ϵ,∞})_{r}^{2} + \frac{1}{r^{2}} (jk)^{2}(u_{ϵ,∞})^{2} + |∇_{x_{3}}u_{ϵ,∞}|^{2}.

By Lemma 2.1 we have

∫_{Ω_{ϵ}^{2jk}} u_{ϵ,∞}^{2} dx_{1} x_{2} dx_{3} ≤ \frac{(diam(Ω_{ϵ}^{2jk}))^{2}}{3} ∫_{Ω_{ϵ}^{2jk}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}.

Thus,

∫_{Ω_{ϵ}} |∇z|^{2} dx_{1} x_{2} dx_{3}
= k ∫_{Ω_{ϵ}^{2jk}} |∇ω|^{2} dx_{1} x_{2} dx_{3}
≤ k ∫_{Ω_{ϵ}^{2jk}}[(1 + \frac{16(jk)^{2}}{27} ϵ^{2})(u_{ϵ,∞})_{r}^{2} + |∇_{x_{3}}u_{ϵ,∞}|^{2}] dx_{1} x_{2} dx_{3}
≤ 2k ∫_{Ω_{ϵ}^{2jk}}((u_{ϵ,∞})_{r}^{2} + |∇_{x_{3}}u_{ϵ,∞}|^{2}) dx_{1} x_{2} dx_{3}  (2)
= 2k ∫_{Ω_{ϵ}^{2jk}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}

provided that ϵ < 3\sqrt{3}/(4jk). By Lemma 2.2 we can find α > 0 such that γ(αz) = 0. Let D = \{(r, θ, x_{3}) : u_{ϵ,∞}(r, |x_{3}|) > ρ, θ ∈ [\frac{π}{4jk} , \frac{π}{2jk}]\}. Suppose that α > \sqrt{2}. Then for (r, θ, x_{3}) ∈ D we have α sin (jkθ) > 2. This, the fact that tf(t) is bounded from below, say by E, and Lemma 2.3 imply

∫_{Ω_{ϵ}} αzf(αz) dx_{1} x_{2} dx_{3}

= k ∫_{Ω_{ϵ}^{2jk}} α sin (jkθ)u_{ϵ,∞}f(α sin (jkθ)u_{ϵ,∞}) dx_{1} x_{2} dx_{3}
≥ k (E|Ω_{ϵ}^{2jk}| + Cα^{2/m} ∫_{D} u_{ϵ,∞}f(u_{ϵ,∞}) dx_{1} x_{2} dx_{3})  (3)

= k (E|Ω_{ϵ}^{2jk}| + Cα^{2/m}(∫_{u_{ϵ,∞}>ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3})(∫_{π/(4jk)}^{π/(2jk)} dθ))

= kE|Ω_{ϵ}^{2jk}| + k \frac{C}{4} α^{2/m}(∫_{u_{ϵ,∞}>ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3})(∫_{0}^{π/(jk)} dθ) ,

where |Ω_{ϵ}| denotes the volume of Ω_{ϵ}.

Now, let ϵ < 1/4. Then K = inf_{u∈S(1/4,k)} J(u) ≤ inf_{u∈S(ϵ,k)} J(u). Let u_{ϵ,∞} also denote the zero extension of u_{ϵ,∞} to all of Ω_{1/4}. Then u_{ϵ,∞} ∈ S(1/4, k). Thus, J(u_{ϵ,∞}) ≥ K and consequently

∫_{Ω_{ϵ}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3} ≥ 2K + 2M|Ω_{ϵ}|  (4)

where M = inf {F (t) : t ∈ ℝ}. If we choose ϵ so that |Ω_{ϵ}| < K/(−2M) then (4) implies

∫_{Ω_{ϵ}} |∇u_{ϵ,∞}|^{2}dx_{1} dx_{2} dx_{3} ≥ K  (5)

and this leads to

∫_{u_{ϵ,∞}>ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}

= ∫_{u_{ϵ,∞}≥0} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3} − ∫_{u_{ϵ,∞}≤ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}

= [1 − \frac{∫_{u_{ϵ,∞}≤ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}}{∫_{u_{ϵ,∞}≥0} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}} ] ∫_{u_{ϵ,∞}≥0} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}

= [1 − \frac{2π ∫_{u_{ϵ,∞}≤ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}}{∫_{Ω_{ϵ}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}} ] ∫_{u_{ϵ,∞}≥0} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}

≥ (1 − Cϵ) ∫_{u_{ϵ,∞}≥0} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}

where the constant C is independent of (ϵ, j, k). By choosing ϵ in such a way that 1 − Cϵ > 1/2 we conclude

∫_{u_{ϵ,∞}>ρ} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3} > \frac{1}{2} ∫_{Ω_{ϵ}} u_{ϵ,∞}f(u_{ϵ,∞})r dr dx_{3}.

This reduces (3) to

∫_{Ω_{ϵ}} αzf(αz) dx_{1} dx_{2} dx_{3} ≥ kE|Ω_{ϵ}^{2jk}| + kCα^{2/m} ∫_{Ω_{ϵ}^{2jk}} |∇u_{ϵ,∞}|^{2} dx_{1} dx_{2} dx_{3}. (6)

On the other hand, using (2) we have

∫_{Ω_{ϵ}} αzf(αz) dx_{1} x_{2} dx_{3} = α^{2} ∫_{Ω_{ϵ}} |∇z|^{2} dx_{1} x_{2} dx_{3}
≤ 2kα^{2} ∫_{Ω_{ϵ}^{2jk}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}.  (7)

Combining (6) and (7) to obtain

(Cα^{2/m} − 2α^{2}) ∫_{Ω_{ϵ}^{2jk}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3} ≤ \frac{Cϵ}{jk}

Hence,

Cα^{2/m} − 2α^{2} ≤ \frac{Cϵ/(jk)}{∫_{Ω_{ϵ}^{2jk}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}}
≤ \frac{Cϵ}{∫_{Ω_{ϵ}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}} ≤ M_{1} ,

where we have used (5). But the function g(α) = Cα^{2/m} −2α^{2} satisfies g(0) = 0 and lim_{α→∞} g(α) = +∞. Thus, if g(α) ≤ M_{1} then there is a constant K′ such that α ≤ K′. By letting

α ≤ max \{\sqrt{2}, K′\} ≡ K′′  (8)

we conclude that α is bounded. Let w = αz ∈ S(ϵ, k). Since F (t) is bounded from below, say by M, then (8) and (2) imply

J(w) = ∫_{Ω_{ϵ}} (\frac{|∇w|^{2}}{2} − F (w)) dx_{1} x_{2} dx_{3}
= ∫_{Ω_{ϵ}} \frac{α^{2}|∇z|^{2}}{2} dx_{1} x_{2} dx_{3} − ∫_{Ω_{ϵ}} F (w) dx_{1} x_{2} dx_{3}
≤ 2(K′′)^{2}k ∫_{Ω_{ϵ}^{2jk}} \frac{|∇u_{ϵ,∞}|^{2}}{2} dx_{1} x_{2} dx_{3} − kM|Ω_{ϵ}^{2jk}|
≤ 2\frac{(K′′)^{2}}{j} ∫_{Ω_{ϵ}} \frac{|u_{ϵ,∞}|^{2}}{2} dx_{1} x_{2} dx_{3} + \frac{kCϵ}{jk}
≤ \frac{(K′′)^{2}}{j} ∫_{Ω_{ϵ}} \frac{|∇u_{ϵ,∞}|^{2}}{2} dx_{1} x_{2} dx_{3} + \frac{1}{j} ∫_{Ω_{ϵ}} \frac{|∇u_{ϵ,∞}|^{2}}{2} dx_{1} x_{2} dx_{3}
≤ \frac{C}{j} ∫_{Ω_{ϵ}} \frac{|∇u_{ϵ,∞}|^{2}}{2} dx_{1} x_{2} dx_{3}  (9)

where we have used (4) with ϵ chosen in such a way that Cϵ < K and |Ω_{ϵ}| < K/(−2M). We claim that there exists a constant C such that

∫_{Ω_{ϵ}} \frac{|∇u_{ϵ,∞}|^{2}}{2} dx_{1} x_{2} dx_{3} ≤ CJ(u_{ϵ,∞}).

Indeed, since γ(u_{ϵ,∞}) = 0 and by (C5) we have

∫_{Ω_{ϵ}} u_{ϵ,∞}f(u_{ϵ,∞}) dx_{1} x_{2} dx_{3}
= ∫_{Ω_{ϵ}} |∇u_{ϵ,∞}|^{2} dx_{1} x_{2} dx_{3}
= ∫_{Ω_{ϵ}}(|∇u_{ϵ,∞}|^{2} − 2F (u_{ϵ,∞})) dx_{1} x_{2} dx_{3} + 2 ∫_{Ω_{ϵ}} F (u_{ϵ,∞}) dx_{1} x_{2} dx_{3}
≤ 2(J(u_{ϵ,∞}) + \frac{m}{2} ∫_{Ω_{ϵ}} u_{ϵ,∞}f(u_{ϵ,∞}) dx_{1} x_{2} dx_{3} + C|Ω_{ϵ}|).

It follows that

(1 − m) ∫_{Ω_{ϵ}} u_{ϵ,∞}f(u_{ϵ,∞}) dx_{1} x_{2} dx_{3} ≤ 2(J(u_{ϵ,∞}) + C|Ω_{ϵ}|).  (10)

On the other hand, using (4) and (C5) we have

J(u_{ϵ,∞}) = ∫_{Ω_{ϵ}}(\frac{1}{2}|∇u_{ϵ,∞}|^{2} − F (u_{ϵ,∞})) dx_{1} x_{2} dx_{3}
≥ (\frac{1 − m}{2}) ∫_{Ω_{ϵ}} u_{ϵ,∞}f(u_{ϵ,∞}) dx_{1} x_{2} dx_{3} − C′|Ω_{ϵ}|
≥ (\frac{1 − m}{2})K − C′|Ω_{ϵ}|.

By choosing ϵ in such a way that C′|Ω_{ϵ}| < \frac{1}{4}(1 − m)K, we see that J(u_{ϵ,∞}) > (\frac{1−m}{4})K. Now, we choose ϵ such that C|Ω_{ϵ}| < (\frac{1−m}{4})K. Using this in (10) we obtain

∫_{Ω_{ϵ}} u_{ϵ,∞}f(u_{ϵ,∞}) dx_{1} x_{2} dx_{3} ≤ CJ(u_{ϵ,∞}).

Also, using this in (9) we obtain

J(w) ≤ \frac{C}{j} J(u_{ϵ,∞}).

Choosing j such that j > 2C we obtain J(w) < J(u_{ϵ,∞}) and this concludes the present proof. ♢

To prove Lemma 1.5 we need

Lemma 2.4 Let v ∈ H_{0}^{1}(Ω_{ϵ}). Then the function

P_{v}(λ) = \frac{λvf(λv)}{2} − F (λv)

is increasing on (0, ∞).

Proof. Differentiating P − v with respect to λ we find

P′_{v}(λ) = \frac{λv^{2}}{2} (f′(λv) − \frac{f(λv)}{λv}) > 0 ,

where we have used (C2). This completes the proof ♢ Proof of Lemma 1.5: Fix k and n. Let 0 < ϵ < ϵ_{1}(kn). Lemma 1.2 guarantees the existence of a minimizer u_{ϵ,kn} of J on S(ϵ, kn). From Lemma 1.3 we see that u_{ϵ,kn} is a solution to (1). Furthermore, from Lemma 1.4 we know that u_{ϵ,kn} is nonradial. Now, by the regularity theory of elliptic equations we know that u_{ϵ,kn} is a C^{2} function. Let x = (r, θ) be the polar coordinates of x ∈ ℝ^{2} and write u = u_{ϵ,kn}(r, θ, |x_{3}|). Then

∫_{Ω_{ϵ}} |∇u|^{2} dx_{1} x_{2} dx_{3} = ∫_{(r,|x_{3}|)} ∫_{0}^{2π}(u_{r}^{2} + \frac{1}{r^{2}} u_{θ}^{2} + |∇_{x_{3}}u|^{2})r dr dθ dx_{3}

and

∫_{Ω_{ϵ}} F (u) dx_{1} x_{2} dx_{3} = ∫_{(r,|x_{3}|)} ∫_{0}^{2π} F (u)r dr dθ dx_{3} .

Define the function

v(r, θ, |x_{3}|) = u(r, \frac{θ}{n}, |x_{3}|), 0 ≤ θ ≤ 2π.

Then

v(r, θ + \frac{2π}{k} , |x_{3}|) = u(r, \frac{θ}{n} + \frac{2π}{kn}, |x_{3}|)
= u(r, \frac{θ}{n}, |x_{3}|)
= v(r, θ, |x_{3}|).

Alfonso Castro & Marcel B. Finan 29 It follows that v ∈ H(ϵ, k). On the other hand, it is easy to check that the following equalities hold

v_{r}(r, θ, |x_{3}|) = u_{r}(r, \frac{θ}{n}, |x_{3}|) ,

v_{θ}(r, θ, |x_{3}|) = \frac{1}{n}u_{θ}(r, \frac{θ}{n}, |x_{3}|) ,

∇_{x_{3}}v(r, θ, |x_{3}|) = ∇_{x_{3}}u(r, \frac{θ}{n}, |x_{3}|).

Therefore,

∫_{Ω_{ϵ}} |∇v|^{2} dx_{1} x_{2} dx_{3} = k ∫_{(r,|x_{3}|)} ∫_{0}^{2π/k}(u_{r}^{2}(r, \frac{θ}{n}, |x_{3}|) + \frac{1}{r^{2}n^{2}} u_{θ}^{2}(r, \frac{θ}{n}, |x_{3}|)

+|∇_{x_{3}}u(r, \frac{θ}{n}, |x_{3}|)|^{2})r dr dθ dx_{3}

= kn ∫0^{\frac{2π}{nk}} (u_{r}^{2}(r, θ, |x_{3}|) + \frac{1}{r^{2}n^{4}} u_{θ}^{2}(r, θ, |x_{3}|)

+|∇_{x_{3}}u(r, θ, |x_{3}|)|^{2})r dr dθ dx_{3}

= ∫_{(r,|x_{3}|)} ∫_{0}^{2π}(u_{r}^{2}(r, θ, |x_{3}|) + \frac{1}{r^{2}n^{4}} u_{θ}^{2}(r, θ, |x_{3}|)

+|∇_{x_{3}}u(r, θ, |x_{3}|)|^{2})r dr dθ dx_{3}.

Also

∫_{Ω_{ϵ}} F (v) dx_{1} x_{2} dx_{3} = k ∫_{(r,|x_{3}|)} ∫_{0}^{2π/k} F (u(r, \frac{θ}{n}, |x_{3}|))r dr dθ dx_{3}
= ∫_{(r,|x_{3}|)} ∫_{0}^{2π} F (u(r, θ, |x_{3}|))r dr dθ dx_{3} .

Since u does not belong to S(ϵ, ∞) we have

∫_{(r,|x_{3}|)} ∫_{0}^{2π} u_{θ}^{2}(r, θ, |x_{3}|)r dr dθdx_{3} > 0 .

It follows that

γ(v) = ∫_{Ω_{ϵ}}(|∇v|^{2} − vf(v)) dx_{1} x_{2} dx_{3}
= ∫_{(r,|x_{3}|)} ∫_{0}^{2π}(u_{r}^{2}(r, θ, |x_{3}|) + \frac{1}{r^{2}n^{4}} u_{θ}^{2}(r, θ, |x_{3}|)
+|∇_{x_{3}}u(r, θ, |x_{3}|)|^{2} − uf(u))r dr dθ dx_{3}
< ∫_{(r,|x_{3}|)} ∫_{0}^{2π}(u_{r}^{2}(r, θ, |x_{3}|) + \frac{1}{r^{2}} u_{θ}^{2}(r, θ, |x_{3}|)
+|∇_{x_{3}}u(r, θ, |x_{3}|)|^{2} − uf(u))r dr dθ dx_{3}
= ∫_{Ω_{ϵ}}(|∇u|^{2} − uf(u)) dx_{1} x_{2} dx_{3} = 0.

This yields

∫_{Ω_{ϵ}} |∇v|^{2} dx_{1} x_{2} dx_{3} < ∫_{Ω_{ϵ}} vf(v) dx_{1} x_{2} dx_{3} .  (11)

Now, by Lemma 2.2 we can find 0 < α < 1 such that αv ∈ S(ϵ, k). Let w = αv ∈ S(ϵ, k). Using Lemma 2.4 and the definition of j_{k}^{ϵ} we have

j_{k}^{ϵ} ≤ J(w) = J(αv) = P_{v}(α)
< P_{v}(1)
= ∫_{Ω_{ϵ}}(\frac{1}{2}vf(v) − F (v)) dx_{1} x_{2} dx_{3}
= ∫_{Ω_{ϵ}}(\frac{1}{2}uf(u) − F (u)) dx_{1} x_{2} dx_{3}
= J(u) = j_{kn}^{ϵ}.

Putting together all the arguments above we conclude a proof of the lemma ♢

3 Proof of Main Theorem

For any integer k ≥ 1, according to Lemma 1.4 there exists ϵ_{1}(2^{k}) such that if 0 < ϵ < ϵ_{1}(2^{k}) then

j_{2^{k}}^{ϵ} < j_{∞}^{ϵ}.

It follows from Lemma 1.5 that

j_{2}^{ϵ} < j_{2^{2}}^{ϵ} < ... < j_{2^{k}}^{ϵ} < j_{∞}^{ϵ}.  (12)

According to Lemma 1.2, there exists u_{ϵ,i} ∈ S(ϵ, i), i = 1, ..., k, such that

j_{2^{i}}^{ϵ} = J(u_{ϵ,i}).

Moreover, by Lemma 1.3 u_{ϵ,i} is a solution of Problem 1, i = 1, …, k. By (12), {u_{ϵ,i}}_{i=1}^{k}, are nonrotationally equivalent and nonradial. This completes the proof of the theorem. ♢

References

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critical point, and existence of sign-changing solutions to elliptic boundary value problems, Electron. J. Differential Equations, Vol. 1998 (1998), N0.2, 1–18.

[5] A. Castro and M.B. Finan, Existence of many sign-changing nonradial solutions for a superlinear Dirichlet problem on thin annuli, Topological Methods in Nonlinear Analysis Vol. 13, No.2 (1999), 273–279

[6] C.V. Coffman, A nonlinear boundary value problem with many positive solutions, J. Differential Equations 54 (1984), 429–437.

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[8] S.S. Lin, Existence of many positive nonradial solutions for nonlinear elliptic equations on an annulus, J. Differential Equations 103 (1993), 338–349.

[9] M. Willem, Minimax Theorems, Birkhäuser, 1996.

Alfonso Castro Division of Mathematics and Statistics University of Texas at San Antonio San Antonio, TX 78249 USA. e-mail: castro@math.utsa.edu

Marcel B. Finan Department of Mathematics The University of Texas at Austin Austin, TX 78712 USA. e-mail: mbfinan@math.utexas.edu