Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 10, 2003, pp 101–107. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

A sign-changing solution for a superlinear Dirichlet problem, II ^{∗}

Alfonso Castro, Pavel Drábek, & John M. Neuberger

Abstract

In previous work by Castro, Cossio, and Neuberger [2], it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is less than the first eigenvalue of −Δ with zero Dirichlet boundry condition. One of these solutions changes sign exactly-once and the other two are of one sign. In this paper we show that when this derivative is between the k-th and k + 1-st eigenvalues there still exists a solution which changes sign at most k times. In particular, when k = 1 the sign-changing exactly-once solution persists although one-sign solutions no longer exist.

1 Introduction

Let Ω be a smooth bounded region in ℝ^{N}, Δ the Laplacian operator, and f ∈ C^{1}(ℝ, ℝ) such that f(0) = 0. In this paper we study the boundary-value problem

\begin{matrix}Δu + f(u) = 0 in Ω \\ u = 0 in ∂Ω. \end{matrix}  (1.1)

We assume that there exist constants A > 0 and p ∈ (1, \frac{N+2}{N−2}) such that |f′(u)| ≤ A(|u|^{p−1} + 1) for all u ∈ ℝ. Hence f is subcritical, i.e., there exists B > 0 such that |f(u)| ≤ B(|u|^{p} + 1). Also, we assume that there exists m ∈ (0, 1) and η > 0 such that

muf(u) ≥ 2F (u)  (1.2)

for |u| > η, where F (u) = ∫_{0}^{u} f(s) ds. Finally, we make the assumption that f satisfies

f′(u) > \frac{f(u)}{u} for u ≠ 0, and lim_{|u|→∞} \frac{f(u)}{u} = ∞ (f is superlinear). (1.3)

∗Mathematics Subject Classifications: 35J20, 35J25, 35J60. Key words: Dirichlet problem, superlinear, subcritical, sign-changing solution, deformation lemma. \begin{matrix}c \\ ◯ \end{matrix}2003 Southwest Texas State University. Published February 28, 2003. P. Drabek was partially supported by Ministry of Education of the Czech Republic, MSM 235200001. J. Neuberger was partially supported by the National Science Foundation DMS-0074326.

Let H be the Sobolev space H_{0}^{1,2}(Ω) with inner product ⟨u, v⟩ = ∫_{Ω} ∇u · ∇v dζ (see [1] or [8]). Let 0 < λ_{1} < λ_{2} ≤ λ_{3} ≤ ⋯ be the eigenvalues of −Δ with zero Dirichlet boundary condition in Ω. We let {ϕ_{1}, ϕ_{2}, … } denote a complete orthonormal set in L^{2}(Ω) of eigenfunctions corresponding to the latter eigenvalues.

Our main result is as follows.

Theorem 1 If f′(0) ∈ [λ_{k}, λ_{k+1}) then (1.1) has a solution w which changes sign at most k times, i.e., Ω − w^{−1}{0} consists of at most k + 1 non-empty connected sets.

Corollary 2 If f′(0) ∈ [λ_{1}, λ_{2}) then (1.1) has a solution w which changes sign exactly once.

Our proofs here combine Lyapunov-Schmidt reduction arguments [5], the mountain pass lemma [11], Sard’s Lemma [12], and the index of critical points of mountain pass type [6].

To the best of our knowledge, [2] was the first to establish the existence of a sign-changing solution to (1.1) for a general region in the superlinear case where f′(0) < λ_{1}. The proofs in [2] are based on the study of the Nehari manifold

S = {u ∈ H : u ≠ 0, ∫_{Ω}(∥∇u∥^{2} − uf(u)) dζ = 0}.

Unlike the work in [2], where S is a differentible manifold homeomorphic to the unit sphere and bounded away from 0, here 0 is a limit point of S. Also S ∪ 0 has a singularity at 0. The intersection of S with planes spanned by {ϕ_{1}, ϕ_{k}}, k = 2, … is a figure eight. The semipostione result in [10] is another example where a more complicated variational structure is successfully analyzed via our techniques.

For historical remarks concerning the existence of sign changing solutions to semilinear elliptic boundary value problems we refer the reader to [4]. See also [13].

Remark 1 One can easily see that when f′(0) ≥ λ_{1} there can be no one signed solutions. In fact suppose to the contrary that f′(0) ≥ λ_{1} and that u is (for example) a positive solution. Let ϕ_{1} be a positive eigenfunction corresponding to λ_{1}. Then, by multiplying (1.1) by ϕ_{1} and integrating we obtain

\begin{aligned}∫_{Ω}\{Δu + f(u)\}ϕ_{1} dζ & = ∫_{Ω}\{uΔϕ_{1} + f(u)ϕ_{1}\} dζ \\ & = ∫_{Ω}\{\frac{f(u)}{u} − λ_{1}\}uϕ_{1} dζ \\ & = ∫_{Ω}\{f′(v) − λ_{1}\}uϕ_{1} dζ \\ & > ∫_{Ω}\{f′(0) − λ_{1}\}uϕ_{1} dζ ≥ 0, \end{aligned}  (1.4)

where we have used the mean value theorem to find v ∈ (0, u).

Remark 2 Theorem 1 and Corollary 2 are also valid when the Dirichlet boundary condition in (1.1) is replaced by a homogeneous boundary condition for which the spectrum of the Laplacian operator consists of isolated eigenvalues of finite multiplicity converging to ∞. This is the case, for example, of the Neumann boundary condition (∂u/∂η)(x) = 0 for region with Lipschitzian boundary.

2 Preliminary Lemmas

Let k be a positive integer and f′(0) ∈ (λ_{k}, λ_{k+1}). We define J : H → ℝ by

J(u) = ∫_{Ω}\{\frac{1}{2}|∇u|^{2} − F (u)\} dζ.

By regularity theory for elliptic boundary value problems [8], u is a solution to (1.1) if and only if u is a critical point of J. Because f is subcritical, J ∈ C^{2}(H, ℝ) (see [11]). The gradient and Hessian of J are given by

J′(u)(v) = ⟨∇J(u), v⟩ = ∫_{Ω}{∇u · ∇v − f(u)v} dζ, for all v ∈ H, (2.1)

and

⟨D^{2}J(u)v, w⟩ = ∫_{Ω}{∇v · ∇w − f′(u)vw} dζ, for all u, v, w ∈ H.  (2.2)

Let X be the linear subspace generated by {ϕ_{1}, …, ϕ_{k}} and Y the subspace of H generated by {ϕ_{k+1}, … }. By orthogonality properties of eigenfunctions H = X ⊕ Y. Since (1.3) implies that f′(t) ≥ f′(0) > λ_{k}, one sees that there exists m_{1} > 0 such that

⟨D^{2}J(u)x, x⟩ ≤ −m_{1}∥x∥^{2} for all u ∈ H, x ∈ X.  (2.3)

Arguing as in Theorem 4 of [5], one sees that there exists a function ψ ∈ C^{1}(Y, X) such that

Ĵ(y) ≡ J(y + ψ(y)) = max_{x∈X} J(x + y) for all y ∈ Y,  (2.4)

and

⟨∇Ĵ(y), v⟩ = ⟨∇J(y + ψ(y)), v⟩ = ∫_{Ω}{∇y · ∇v − f(y + ψ(y))v} dζ, (2.5)

for all y, v ∈ Y. In addition,

ψ(y) is the only critical point of x → J(x + y) for all y ∈ Y.  (2.6)

Thus

y is a critical point of Ĵ if and only if y + ψ(y) is a critical point of J. (2.7)

Although not obvious (see [5]), Ĵ is of class C^{2} in spite of only ψ ∈ C^{1}(Y, X). From (2.5) we see that ∇Ĵ(y) = y + K(y) where K is a compact function of y. Also since ∇Ĵ is a variational vector field of class C^{1} we have dim ker(∇Ĵ)′(y) = dim ker(D^{2}Ĵ(y)) = codim(D^{2}(Ĵ(y))(Y)) = codim(∇J)′(y)(Y) for all y ∈ Y. Thus we may apply Sard’s lemma [12] to conlcude the following lemma.

Lemma 1 There exists {q_{n}} ⊂ Y with q_{n} ↓ 0 as n → ∞ such that if ∇Ĵ(u) = q_{n} then the Hessian D^{2}Ĵ(u) is invertible.

Proof This proof follows immediately from the fact that the regular values of ∇Ĵ is the complement of a set of first category. In partiuclar it is dense. ♢

Let q_{n} be as in the previous lemma. We define J_{n} : Y → ℝ by J_{n}(y) = Ĵ(y) − ⟨q_{n}, y⟩. We note that D^{2}J_{n}(y) = D^{2}Ĵ(y) for each y ∈ Y. Also from (12) of [5]

⟨D^{2}Ĵ(y)h, h⟩ = ⟨D^{2}J(y + ψ(y))(h + ψ′(y)h), h + ψ′(y)h⟩.  (2.8)

Lemma 2 The functional J_{n} has a critical point y_{n} such that the Morse index of D^{2}J(y_{n} + ψ(y_{n})) is less than or equal to k + 1.

proof In order to establish the existence of the critical point u_{n} we prove that J_{n} satisifies the hypotheses of the Mountain Pass Lemma [11]. Since f′(0) < λ_{k+1}, we have ⟨D^{2}J(0)y, y⟩ = J′′(0)(y, y) ≥ (1 − f′(0)/λ_{k+1}) ∫_{Ω} |∇y|^{2} dζ > 0 for y ∈ Y. Thus 0 is a strict local minimum of J restricted to Y. Thus there exist δ, η > 0 such that J(y) ≥ η for all ||y|| = δ. This and (2.4) imply that for ∥y∥ = δ we have

Ĵ(y) ≥ J(y) ≥ η > 0 .  (2.9)

Now taking n sufficiently large so that ∥q_{n}∥ ≤ η/(2δ) we have

J_{n}(y) ≥ η − ∥q_{n}∥δ > η/2 > 0 ,  (2.10)

for ∥y∥ = δ.

Next we note that, since 0 is a critical point of J, ψ(0) = 0. Hence J_{n}(0) = 0. Additionally, since we are assuming f to be superlinear, there exit numbers m_{1} > λ_{k+1} and m_{2} such that

2F (t) ≥ m_{1}t^{2} + m_{2}  (2.11)

for all t ∈ ℝ. Hence

\begin{aligned}J_{n}(tϕ_{k+1}) & = Ĵ(tϕ_{k+1}) − ⟨q_{n}, tϕ_{k+1}⟩ ≤ J(tϕ_{k+1}) − ⟨q_{n}, tϕ_{k+1}⟩ \\ & ≤ (t∥ϕ_{k+1}∥^{2} − m_{1} ∫_{Ω}(tϕ_{k+1})^{2} dζ − m_{2}|Ω|)/2 + t∥q_{n}∥∥ϕ_{k+1}∥ \\ & → −∞ as t → +∞, \end{aligned} (2.12)

where we have used that ∥ϕ_{k+1}∥^{2} = λ_{k+1} ∫_{Ω} ϕ_{k+1}^{2} dζ. For future reference we note that, without loss of generality, we may assume that ∥q_{n}∥ ≤ 1 for all n.

Thus (2.12) implies that

J_{n}(tϕ_{k+1}) ≤ \frac{(λ_{k+1} − m_{1})(−m_{2}|Ω|) − λ_{k+1}∥ϕ_{k+1}∥}{2(λ_{k+1} − m_{1})} ≡ K̃ for all n, t ≥ 0.

(2.13)

From (2.12), for each n, there exists a real number t_{n} > 0 with ∥t_{n}ϕ_{k+1}∥ ≥ 2δ such that Ĵ_{n}(t_{n}ϕ_{k+1}) < 0.

Next we show that J_{n} satisfies the Palais-Smale condition. Suppose that {y_{j}} is a sequence so that {J_{n}(y_{j})} is bounded, say |J_{n}(y_{j})| ≤ M for all j and ∇J_{n}(y_{j}) → 0 as j → ∞. For ease of notation, let u = y_{j} + ψ(y_{j}) and T = \frac{m}{2}uf(u) − F (u). Then

\begin{aligned}M + \frac{m}{2}||u|| & ≥ J_{n}(y_{j}) − \frac{m}{2}[⟨∇J_{n}(y_{j}), y_{j}⟩ + ⟨∇J(y_{j} + ψ(y_{j})), ψ(y_{j})⟩] \\ & = (\frac{1}{2} − \frac{m}{2})||u||^{2} + ∫_{Ω} T dζ − (1 − \frac{m}{2})⟨q_{n}, u⟩ \\ & ≥ (\frac{1}{4} − \frac{m}{4})(||u||^{2} − ∥q_{n}∥^{2}) + M_{1}|Ω|, \end{aligned} (2.14)

where M_{1} ∈ ℝ is a lower bound for T (see (1.2)). The latter inequality implies that {y_{j} + ψ(y_{j})} is bounded. Hence without loss of generality we may assume that {y_{j}} converges weakly to ȳ ∈ Y and that ψ(y_{j}) converges to x̄ ∈ X. Since the imbedding of H in L^{p+1}(Ω) is compact, we may assume that f(y_{j} + ψ(y_{j})) converges strongly in L^{1}(Ω). Since ∇J_{n}(y_{j}) = y_{j} + K(y_{j}) + q_{n} → 0 with K compact, we see that {y_{j}} has a convergent subsequence. This proves that J_{n} satisfies the Palais-Smale condition.

Now by the Mountain Pass Lemma there exists y_{n} ∈ Y such that ∇J_{n}(y_{n}) = 0 and

J_{n}(y_{n}) = inf_{σ∈Σ} [ max_{t∈[0,1]} J_{n}(σ(t))],  (2.15)

where Σ = {σ : [0, 1] → Y; σ is continuous, σ(0) = 0 and σ(1) = t_{n}ϕ_{k+1}}. Thus from (2.10), (2.13) and (2.15) we have

η/2 ≤ J_{n}(y_{n}) ≤ K̃.  (2.16)

In addition, since D^{2}J_{n}(y_{n}) is invertible (see Lemma 1), by Theorem 2 of [6], we may assume that the Morse index of D^{2}J_{n}(y_{n}) is 1. Since D^{2}J(y_{n} + ψ(y_{n})) is negative definite in a subspace of dimension k, namely X, then we conclude that the Morse index of D^{2}J(y_{n} + ψ(y_{n})) is at most k + 1, and this concludes the proof. ♢

3 Proof of Main Theorem

First we consider the case f′(0) > λ_{k}. Let {y_{n}} be as in Lemma 2. By (2.16) one sees that {y_{n} +ψ(y_{n})} is bounded. Thus, without loss of generality, we may assume that {y_{n}} converges weakly to ȳ ∈ Y and {ψ(y_{n})} converges to x̄ ∈ X.

From (2.4) we have 0 = ∇J_{n}(y_{n}) = y_{n} + K(y_{n}) − q_{n} where K is a compact operator. Since, in addition {q_{n}} converges to 0, actually {y_{n}} converges strongly to ȳ. Also since J_{n}(y_{n}) ≥ η/2 > 0, we see that Ĵ(ȳ) ≥ η/2 > 0 and ȳ ≠ 0. Now for v ∈ X one has

\begin{matrix}⟨x̄, v⟩ − ∫_{Ω}\{f(ȳ + x̄)v\} dζ \\ = lim_{n→∞} [⟨ψ(y_{n}), v⟩ − ∫_{Ω}\{vf(y_{n} + ψ(y_{n}))\} dζ − ⟨q_{n}, ψ(y_{n})⟩] \\ = 0. \end{matrix}  (3.1)

Hence x̄ = ψ(ȳ) and ∇J(x̄ + ȳ) = 0. Thus x̄ + ȳ ≠ 0 is a solution to (1.1). Let us see that x̄ + ȳ has at most k + 1 nodal regions. If not, by defining v_{j}, j = 1, …, k+2, as x̄+ȳ on W_{j} and as zero on Ω̄−W_{j}, then from (1.3), (2.1) and (2.2), we see that ⟨D^{2}J(x̄+ȳ)v_{j}, v_{j}⟩ < 0. Since the v′_{j}s are mutually orthogonal then we have that D^{2}J(x̄+ȳ) is negative definite on a k+2-dimensional subspace. By continuity then D^{2}J(y_{n} + ψ(y_{n})) is negative definite on the same k + 2dimensional subspace. This contradicts that the Morse index of D^{2}J(y_{n}+ψ(y_{n})) is less than or equal to k + 1. This contradiction proves that x̄ + ȳ is a solution to (1.1) having at most k + 1 nodal regions.

Finally we consider the case f′(0) = λ_{k}. Let {ϵ_{j}} be a sequence of positive numbers converging to 0. Without loss of generality we may assume that ϵ_{j} < (λ_{k+1} −λ_{k})/2 for all positive integers j. By our previous arguments, there exists a sequence {u_{j} = x̄_{j} + ȳ_{j}} of functions in H that satisfy

\begin{matrix}Δu_{j} + ϵ_{j}u_{j} + f(u_{j}) = 0 in Ω \\ u_{j} = 0 in ∂Ω. \end{matrix}  (3.2)

In addition, each x_{j} + y_{j} is the limit a of a sequence {x_{n,j} + y_{n,j}} with each x_{n,j} + y_{n,j} satisfying (2.16). Hence, by continuity η/2 ≤ J(x̄_{j} + ȳ_{j}) ≤ K̃ and

∫_{Ω}{∇v · ∇v − (f′(u_{j}) + ϵ_{j})v^{2}} dζ

defines a quadratic form of Morse index at most k + 1. Arguing as above one sees that x̄_{j} + ȳ_{j} has a convergent subsequence with limit x̄ + ȳ. The function x̄ + ȳ is a solution to (1.1) and, as in the case f′(0) > λ_{k}, it has at most k + 1 nodal regions. This concludes the proof of our main theorem; the corollary is obvious.

References

[1] R. Adams, Sobolev Spaces, New York: Academic Press (1975).

[2] A. Castro, J. Cossio and J. M. Neuberger, A Sign-Changing Solution for a Superlinear Dirichlet Problem, Rocky Mountain J. of Math., 27, No. 4 (1997), pp. 1041-1053.

[3] A. Castro, J. Cossio and J. M. Neuberger, On Multiple Solutions of a Nonlinear Dirichlet Problem, Nonlinear Analysis TMA, 30, No. 6 (1997), pp. 3657-3662.

[4] A. Castro, J. Cossio and J.. M. Neuberger, A Minmax Principle, Index of the Critical Point, and Existence of Sign-Changing Solutions to Elliptic Boundary Value Problems, Electronic J. of Diff. Eq., Vol. 1998 (1998), No. 2, pp. 1-18.

[5] A. Castro, and A. C. Lazer, Critical Point Theory and the Number of Solutions of a Nonlinear Dirichlet Problem, Annli di Mat. Pura ed Applicata (IV), Vol. CXX (1979), pp. 113-137.

[6] H. Hofer, The Topological Degree at a Critical Point of Mountain Pass Type, Proceedings of Symposia in Pure Mathematics, 45 Part I (1986).

[7] D. Kinderlehrer and G. Stampacchia, Introduction to Variational Inequalities and Their Applications, New York: Academic Press (1979).

[8] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Berlin, New York: Springer-Verlag (1983).

[9] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, Cambridge University Press (1993).

[10] John M. Neuberger, A Sign-Changing Solution for a Superlinear Dirichlet Problem with a Reaction Term Nonzero at Zero, Nonlinear Analysis, 33 (1998), pp. 427-441.

[11] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Regional Conference Series in Mathematics, 65, Providence,R.I.: AMS (1986).

[12] S. Smale, An Infinite Dimensional Version of Sard’s Theorem, Amer. J. Math., 87 (1965), pp. 861-866.

[13] Z. Q. Wang, On a Superlinear Elliptic Equation, Ann. Inst. H. Poincare Analyse Non Lineaire 8 (1991) 43-57.

Alfonso Castro Division of Mathematics and Statistics, University of Texas at San Antonio, San Antonio, TX 78249-0664, USA E-mail: acastro@utsa.edu Pavel Drabek Department of Mathematics, University of West Bohemia, 306 14 Pilsen, Czech Republic. E-mail address: pdrabek@kma.zcu.cz John M. Neuberger Department of Mathematics, Northern Arizona University, Flagstaff, AZ 86011-5717 USA. E-mail address: John.Neuberger@nau.edu