2006 International Conference in Honor of Jacqueline Fleckinger. Electronic Journal of Differential Equations, Conference 16, 2007, pp. 81–93. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

SOME REMARKS ON INFINITE-DIMENSIONAL NONLINEAR ELLIPTIC PROBLEMS

PHILIPPE CLÉMENT, MARTA GARCÍA-HUIDOBRO, RAÚL F. MANÁSEVICH

Dedicated to Jacqueline Fleckinger on the occasion of an international conference in her honor

Abstract. We discuss some nonlinear problems associated with an infinite dimensional operator L defined on a real separable Hilbert space H. As the operator L we choose the Ornstein-Uhlenbeck operator induced by a centered Gaussian measure μ with covariance operator Q.

1. Introduction

The goal of this note is to present some results for nonlinear problems associated with an infinite dimensional operator L defined on a real separable Hilbert space H. As the operator L we choose the Ornstein-Uhlenbeck operator induced by a centered Gaussian measure μ with covariance operator Q (see [8]).

In the first part we consider existence and uniqueness of solutions for a problem of the form

−Lu + β(u) = f,  (1.1)

where β satisfies

(H1) β is a strictly increasing homeomorphism of ℝ onto ℝ, β(0) = 0, and f ∈ L^{2}(H, μ) is given. As a consequence of the existence part we can show that the operator L(β^{−1}), with an appropriate domain, has an m-dissipative closure in L^{1}(H, μ). Thus, in view of the Crandall-Liggett Theorem, see [7] (and also [6]), it generates a nonlinear contraction semigroup on the closure of its domain in L^{1}(H, μ).

In the second part we make the additional assumption that β is odd and we consider the nonlinear eigenvalue problem

−Lu + β(u) = λu, λ ≥ 0,  (1.2)

2000 Mathematics Subject Classification. 35J65, 35J25.

Key words and phrases. Hilbert space; Ornstein-Uhlenbeck operator;

nonlinear elliptic problems.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published May 15, 2007.

Ph. Clément was supported by grant 7150117 from FONDECYT; M. García-H. by

grant 1030593 from FONDECYT; R. Manásevich by grant P04-066-F from Fondap

Matemáticas Aplicadas and Milenio.

where u ∈ L^{2}(H, μ), ∥u∥_{L^{2}(H,μ)} = R, with R > 0 given.

By using results in [4] and [9], we obtain the existence of an infinite sequence {(λ_{n}, u_{n})}_{n∈ℕ} of solutions to (1.2) with λ_{n} → ∞ as n → ∞. This implies the existence of infinitely many solution pairs (λ, u) with non constant u. Moreover, we discuss the existence of solutions with nonnegative and non constant u.

2. Preliminaries

In this section we establish the notation that we will use throughout this work. Most of it is taken from [8] and we refer the reader to this book. H will denote a finite or infinite dimensional real separable Hilbert space with inner product ⟨·, ·⟩ and norm | · |. Throughout the paper μ = N_{Q} will denote the centered Gaussian measure on H with covariance Q, (see [8, page 12]), where Q denotes a positive symmetric operator of trace class in H with Ker(Q) = {0}. Also, {e_{k}}_{k∈ℕ} will denote a complete orthonormal system of eigenvectors of Q with corresponding eigenvalues {γ_{k}}_{k∈ℕ} satisfying

0 < γ_{k+1} ≤ γ_{k}.  (2.1)

We recall here that the Ornstein-Uhlenbeck semigroup “associated with μ” is given by

R_{t}φ(x) = ∫_{H} φ(e^{tA}x + y)N_{Q_{t}}(dy), x ∈ H, t > 0,

and φ ∈ B_{b}(H) (Borel bounded functions on H). Here A = −\frac{1}{2}Q^{−1} and

Q_{t}x = ∫_{0}^{t} e^{2sA}xds = Q(I − e^{2tA})x, x ∈ H, t > 0.

As a consequence of [8, Proposition 10.22], R_{t} can be uniquely extended to a strongly continuous contraction semigroup in L^{2}(H, μ), which we still denote by R_{t}, and μ is the unique invariant measure of R_{t} and for x ∈ H,

lim_{t→∞} R_{t}φ(x) = ∫_{H} φ(y)dμ(y) = \overline{φ}.

Moreover, from [8, Th5.8], R_{t} can be uniquely extended to a strongly continuous positive contraction semigroup in L^{p}(H, μ) for all 1 ≤ p < ∞.

We shall denote by L_{p} the infinitesimal generator of R_{t} in L^{p}(H, μ). In particular, L_{1} is m-dissipative in L^{1}(H, μ) hence it satisfies

∫_{H}(L_{1}u)(x) sgn(u(x))dμ ≤ 0, for every u ∈ D(L_{1}),  (2.2)

see e.g. [3, Lemma 2], where we have used the notation

sgn(t) = \begin{cases}1 & t > 0, \\ 0 & t = 0, \\ −1 & t < 0. \end{cases}

Moreover, −L_{2} is a nonnegative self adjoint operator in L^{2}(H, μ) with domain

D(L_{2}) = {u ∈ W^{2,2}(H, μ) : ∫_{H} |(−A)^{1/2}Du|^{2}dμ < ∞},  (2.3)

see [8, Propositions 10.22 and 10.34] and

L_{2}φ(x) = \frac{1}{2}Tr[D^{2}φ](x) + ⟨x, ADφ(x)⟩,  (2.4)

where φ ∈ E_{A}(H), which is defined to be the linear span in C_{b}(H) (continuous bounded functions in H) of real and imaginary parts of φ_{h}, where φ_{h}(x) = e^{i⟨h,x⟩}, h ∈ D(A), and D, D^{2} are the differential operators introduced in [8, Proposition 10.3 and 10.32]. We also introduce E(H) as the linear span in C_{b}(H) of real and imaginary parts of φ_{h}, where φ_{h}(x) = e^{i⟨h,x⟩}, and now h ∈ H. Finally we note also that the null space N(L_{p}) = {const.}, 1 ≤ p < ∞.

We also consider the Dirichlet form a : W^{1,2}(H, μ) × W^{1,2}(H, μ) → ℝ defined by

a(φ, ψ) = \frac{1}{2} ∫_{H}⟨Dφ, Dψ⟩dμ.

The linear space W^{1,2}(H, μ) endowed with the inner product

⟨φ, ψ⟩_{W^{1,2}(H,μ)} = ⟨φ, ψ⟩_{L^{2}(H,μ)} + 2a(φ, ψ)

is a real separable Hilbert space with

W^{1,2}(H, μ) ↪ L^{2}(H, μ) compact,  (2.5)

see [8, Theorem 10.16]. Finally, we recall that

a(φ, ψ) = − ∫_{H}⟨L_{2}φ, ψ⟩dμ  (2.6)

for all φ ∈ D(L_{2}), and all ψ ∈ W^{1,2}(H, μ), see [8, Section 10.4].

Remark 2.1. We want to note that in this work the operator L_{2} is defined as the generator of the semigroup R_{t} in L^{2}(H, μ), while in [8] the operator L_{2} is defined on page 151 via the Lax-Milgram Theorem. In view of [8, Proposition 10.22 (iv)], they are the same.

3. An infinite dimensional porous media type operator

The aim of this section is to construct an infinite dimensional nonlinear second order elliptic operator which is of porous media type Δ(β^{−1}), following the approach of [5]

Let β satisfy (H1) and consider problem (1.1).

Proposition 3.1. (a) For every f ∈ L^{2}(H, μ) there exists a unique u ∈ D(L_{2}) such that β(u) ∈ L^{2}(H, μ) and u satisfies (1.1) with L = L_{2}.

(b) If

(H2) β(u) = εu+γ(u) for some ε > 0 and some continuous monotone increasing

function γ : ℝ → ℝ, γ(0) = 0, then for any f ∈ L^{1}(H, μ) there exists a unique u ∈ D(L_{1}) with β(u) ∈ L^{1}(H, μ) satisfying (1.1) with L = L_{1}.

Proof. We start by proving (a). Set A := −L_{2}, Bu(x) := β(u(x)), where

D(B) = {u ∈ L^{2}(H, μ) : β(u) ∈ L^{2}(H, μ)}

and write (1.1) as

Au + Bu = f, f ∈ L^{2}(H, μ).

We claim that A is maximal monotone and that it is the subdifferential of the convex l.s.c functional J_{a} : L^{2}(H, μ) ↦ [0, ∞] defined by

J_{a}(φ) = \begin{cases}a(φ, φ), & φ ∈ W^{1,2}(H, μ), \\ +∞ & otherwise. \end{cases}  (3.1)

Indeed, for u ∈ D(L_{2}) and h ∈ W^{1,2}(H, μ), by (2.6), we have that

J_{a}(u + h) = J_{a}(u) + ∫_{H}⟨Du, Dh⟩dμ + J_{a}(h)
≥ J_{a}(u) − ∫_{H}⟨L_{2}u, h⟩dμ,

which implies that u ∈ D(∂J_{a}) and −L_{2}u ∈ ∂J_{a}(u). Note that since J_{a} is convex, it follows that ∂J_{a} is monotone, moreover, since −L_{2} is nonnegative and selfadjoint in L^{2}(H, μ), it follows that it is maximal monotone. Hence −L_{2} = ∂J_{a} by the maximal monotonicity of −L_{2}. Also, B is the subdifferential of

J_{b}(u) = \begin{cases}∫_{H} b(u)dμ, & if ∫_{H} b(u)dμ < ∞ \\ ∞ & otherwise, \end{cases}

where

b(t) := ∫_{0}^{t} β(s)ds.  (3.2)

Therefore A and B satisfy all the assumptions in [2, Example 1] implying that

Int(R(A + B)) = Int(R(A) + R(B)).

Since R(B) = L^{2}(H, μ), we conclude that R(A + B) = L^{2}(H, μ). Finally, the uniqueness assertion follows from the strict monotonicity of β.

Next we prove (b). In order to achieve this we write (1.1) as

(ε − L_{1})u + γ(u) = f, f ∈ L^{1}(H, μ).  (3.3)

Hence in view of Theorem 1 in [3] it is sufficient to see that the operator A := ε − L_{1} satisfies (I), (II), and (III) in [3]. Since by definition L_{1} generates a linear contraction C_{0} semigroup in L^{1}(H, μ), so does L_{1} −ε = −A, which yields (I). Also, from the dissipativity of L_{1} we have that

ε∥u∥_{L^{1}(H,μ)} ≤ ∥εu − L_{1}∥_{L^{1}(H,μ)} = ∥A∥_{L^{1}(H,μ)},

implying that (III) is also satisfied. Finally we prove (II). Let λ > 0 and f ∈ L^{1}(H, μ). Since the semigroup generated by L_{1} is positive, we have that

(I + λA)^{−1}f ≤ (I + λA)^{−1}f^{+}

and hence

ess sup(I + λA)^{−1}f ≤ ess sup(I + λA)^{−1}f^{+}.  (3.4)

Since L_{p} generates a linear contraction C_{0} semigroup in L^{p}(H, μ) for all 1 ≤ p < ∞, so does L_{p} − ε, hence

∥(I + λA)^{−1}f^{+}∥_{L^{p}(H,μ)} ≤ ∥f^{+}∥_{L^{p}(H,μ)},  (3.5)

provided that f^{+} ∈ L^{p}(H, μ). Assuming f^{+} ∈ L^{∞}(H, μ), by letting p → ∞ in (3.5) we obtain

ess sup(I + λA)^{−1}f^{+} = ∥(I + λA)^{−1}f^{+}∥_{L^{∞}(H,μ)}
≤ ∥f^{+}∥_{L^{∞}(H,μ)} = ess sup f^{+}
= max{0, ess sup f}.

Therefore, using (3.4) we conclude that

ess sup(I + λA)^{−1}f ≤ max{0, ess sup f},

which is exactly assumption (II) in [3]. □

We are now in a position to define a “porous media type” operator, which we denote by L_{ϕ}, where ϕ = β^{−1} in L^{1}(H, μ):

D(L_{Φ}) := {u ∈ L^{1}(H, μ) : ϕ(u) ∈ D(L_{1})},

and for u ∈ D(L_{Φ}) we set

L_{ϕ}u := L_{1}(ϕ(u)).

We have the following result.

Theorem 3.2. (i) The closure of L_{ϕ} is a nonlinear (possibly multivalued) m-

dissipative operator in L^{1}(H, μ). (ii) If β satisfies assumption (H2), then L_{ϕ} is itself a nonlinear m-dissipative

operator in L^{1}(H, μ). (iii) If ϕ ∈ C^{2}(ℝ), then \overline{D(L_{ϕ})} = L^{1}(H, μ).

Remark 3.3. We do not claim that the last two assertions in Theorem 3.2 are optimal.

Proof of Theorem 3.2. (i) We will first prove the dissipativity of L_{ϕ} in L^{1}(H, μ). Let u, v ∈ D(L_{ϕ}) and let ū = ϕ(u), v̄ = ϕ(v). By assumption, ū and v̄ belong to D(L_{1}). In view of the dissipativity of L_{1} in L^{1}(H, μ) we have

∫_{H} L_{1}(ū − v̄) sgn(ū − v̄)dμ ≤ 0,  (3.6)

and in view of (H1),

sgn(u − v) = sgn(ū − v̄).  (3.7)

Hence, replacing (3.7) into (3.6), and using the definition of ū, v̄ we get

∫_{H}(L_{1}(ϕ(u) − ϕ(v)) sgn(u − v)dμ ≤ 0,

which implies the dissipativity of L_{ϕ}. We prove now that R(I − L_{ϕ}) is dense in L^{1}(H, μ). Let f ∈ L^{2}(H, μ). Then by Proposition 3.1 (a), there exists v ∈ D(L_{2}), with β(v) ∈ L^{2}(H, μ), such that

−L_{2}v + β(v) = f,

hence setting u = β(v) we obtain v = ϕ(u) and

u − L_{2}ϕ(u) = f,

hence f ∈ R(I − L_{ϕ}) (since L_{2} ⊂ L_{1}). We conclude that L^{2}(H, μ) ⊆ R(I − λL_{ϕ}) and the claim follows from the density of L^{2}(H, μ) in L^{1}(H, μ).

It is a well known fact that if \overline{L_{ϕ}} denotes the closure of L_{ϕ}, then \overline{L_{ϕ}} is dissipative (possibly multivalued) and R(I − \overline{L_{ϕ}}) is closed, hence equal to L^{1}(H, μ). Therefore \overline{L_{ϕ}} is m-dissipative in L^{1}(H, μ). (ii) It follows from Proposition 3.1 that if β is of the form (H2) then the range

R(I − L_{ϕ}) = L^{1}(H, μ),

hence in this case L_{ϕ} is m-dissipative. (iii) It is sufficient to show that E_{A}(H) ⊆ D(L_{ϕ}), since E_{A}(H) is dense in L^{2}(H, μ). If v ∈ E_{A}(H), then there exists N ≥ 1, h_{1}, h_{2}, …, h_{N}, k_{1}, k_{2}, …, k_{N} ∈ D(A) α_{1}, α_{2}, …, α_{N}, β_{1}, β_{2}, …, β_{N} ∈ ℝ such that

v(x) = ∑_{i=1}^{N}(α_{i} cos⟨h_{i}, x⟩ + β_{i} sin⟨k_{i}, x⟩), x ∈ H.  (3.8)

We will prove that ϕ(v) ∈ D(L_{2}). In view of (2.3), with first verify that ϕ(v) ∈ W^{2,2}(H, μ). Since v ∈ C_{b}(H), we have that ϕ(v), ϕ′(v) and ϕ′′(v) are in C_{b}(H). In particular, ϕ(v) ∈ L^{2}(H, μ).

From the definition of W^{2,2}(H, μ) in [8, Section 10.6, page 161], we need to compute D_{j}D_{ℓ}ϕ(v), j, ℓ ∈ ℕ. Since D_{j}v and D_{ℓ}v are bounded and continuous, from

D_{ℓ}ϕ(v) = ϕ′(v)D_{ℓ}v,

and

D_{j}D_{ℓ}ϕ(v)(x) = ϕ′(v)D_{j}D_{ℓ}v(x) + ϕ′′(v)D_{j}v(x)D_{ℓ}v(x),  (3.9)

we obtain that D_{j}D_{ℓ}ϕ(v) ∈ C_{b}(H) ⊆ L^{2}(H, μ).

Next we show that

∑_{j,ℓ=1}^{∞} ∫_{H} |D_{j}D_{ℓ}ϕ(v)|^{2}dμ < ∞.  (3.10)

From (3.9) it is sufficient to show that

∑_{j,ℓ=1}^{∞} ∫_{H} |D_{j}D_{ℓ}v|^{2}dμ < ∞ and ∑_{j,ℓ=1}^{∞} ∫_{H} |D_{j}v(x)D_{ℓ}v(x)|^{2}dμ < ∞.  (3.11)

Indeed, the first assertion in (3.11) follows from the fact that v ∈ E_{A}(H) ⊆ E(H) ⊆ W^{2,2}(H, μ). For the second one we note that

|D_{j}v(x)| ≤ C ∑_{i=1}^{N}(|⟨h_{i}, e_{j}⟩| + |⟨k_{i}, e_{j}⟩|)  (3.12)

where C is a positive constant depending only on α_{1}, …, α_{N}, β_{1}, …, β_{N}, hence

∑_{j,ℓ=1}^{∞} |D_{j}v(x)D_{ℓ}v(x)|^{2} ≤ 4N^{2}C^{4}(∑_{i=1}^{N} |h_{i}|^{2} + |k_{i}|^{2})^{2},  (3.13)

implying that the second assertion in (3.11) holds and therefore ϕ(v) ∈ W^{2,2}(H, μ). Finally we will prove that

∫_{H} |(−A)^{1/2}Dϕ(v)|^{2}dμ < ∞.  (3.14)

First we prove that Dϕ(v)(x) ∈ D((−A)^{1/2}). Since A = −\frac{1}{2}Q^{−1}, w ∈ H belongs to D((−A)^{1/2}) if and only if

∑_{j=1}^{∞} γ_{j}^{−1}⟨w, e_{j}⟩^{2} < ∞.  (3.15)

Now, Dϕ(v)(x) = ϕ′(v)D_{j}v(x) and |ϕ′(v)| ≤ C_{0} for some positive constant C_{0}, hence from (3.12) we find that

|D_{j}ϕ(v)(x)|^{2} ≤ C_{0}^{2}|D_{j}v(x)|^{2} ≤ 2NC_{0}^{2}C^{2} ∑_{i=1}^{N}(|⟨h_{i}, e_{j}⟩|^{2} + |⟨k_{i}, e_{j}⟩|^{2})

where h_{i}, k_{i} ∈ D(A), i = 1, …, N, that is,

∑_{j=1}^{∞} γ_{j}^{−2}|⟨h_{i}, e_{j}⟩|^{2} < ∞ and ∑_{j=1}^{∞} γ_{j}^{−2}|⟨k_{i}, e_{j}⟩|^{2} < ∞.  (3.16)

Hence from (3.16),

∑_{j=1}^{∞} γ_{j}^{−1}|D_{j}ϕ(v)(x)|^{2} ≤ 2NC_{0}^{2}C^{2} ∑_{i=1}^{N} ∑_{j=1}^{∞} γ_{j}^{−1}(|⟨h_{i}, e_{j}⟩|^{2} + |⟨k_{i}, e_{j}⟩|^{2}) < ∞,

since by (2.1) γ_{j}^{−1} ≤ γ_{j}^{−2} for large j. This implies that Dϕ(v)(x) ∈ D((−A)^{1/2}) for any x ∈ H and

|(−A)^{1/2}Dϕ(v)|^{2}(x) = ∑_{j=1}^{∞}⟨Dϕ(v)(x), (−A)^{1/2}e_{j}⟩^{2}

= \frac{1}{2} ∑_{j=1}^{∞}⟨Dϕ(v)(x), γ_{j}^{−1/2}e_{j}⟩^{2}

= \frac{1}{2} ∑_{j=1}^{∞} γ_{j}^{−1}⟨Dϕ(v)(x), e_{j}⟩^{2},

implying that the integrand in (3.14) is Borel measurable and bounded and thus (3.14) holds. This completes the proof of part (3). □

We end this section by giving some properties of the nonlinear semigroup generated by \overline{L_{ϕ}}.

Proposition 3.4. Let β satisfy (H1) and S_{t} : \overline{D(\overline{L_{ϕ}})} → \overline{D(\overline{L_{ϕ}})} be the nonlinear semigroup generated by \overline{L_{ϕ}}. Then the following hold.

(i) For any c ∈ ℝ, c ∈ D(L_{ϕ}), and S_{t}(c) = c.

(ii) Let f, g ∈ \overline{D(\overline{L_{ϕ}})} such that f ≤ g. Then S_{t}(f) ≤ S_{t}(g) for all t > 0.

(iii) For any f ∈ \overline{D(\overline{L_{ϕ}})},

∫_{H} S_{t}fdμ = ∫_{H} fdμ for all t > 0.

Proof. From Proposition 3.1, for any h > 0 there is a unique u ∈ L^{2}(H, μ) such that

−L_{2}u + \frac{1}{h}β(u) = \frac{1}{h}f,  (3.17)

hence

(I − h\overline{L_{ϕ}}))^{−1}f = β(u).  (3.18)

Proof of (i). If f = c, we have β(u) = c and thus by induction it follows that

(I − h\overline{L_{ϕ}})^{−m}c = c for all m ∈ ℕ,  (3.19)

therefore, for any t > 0 we have

S_{t}(c) = lim_{m→∞}(I − \frac{t}{m}\overline{L_{ϕ}})^{−m}c = c

Proof of (ii). Let now f_{1}, f_{2} ∈ L^{2}(H, μ), with f_{1} ≤ f_{2}, and for h > 0 and ε > 0, and i = 1, 2, let u_{i}^{ε} satisfy

εu_{i}^{ε} − L_{2}u_{i}^{ε} + \frac{1}{h}β(u_{i}^{ε}) = \frac{1}{h}f_{1}.

From [1, Proposition 4.7 (iv) implies (i)] with

φ(u) = \begin{cases}0 & u ≥ 0 \\ +∞ & otherwise, \end{cases}

we obtain u_{1}^{ε} ≤ u_{2}^{ε}. By letting ε → 0 we obtain u_{1} ≤ u_{2} where u_{i} satisfy

−L_{2}u_{i} + \frac{1}{h}β(u_{i}) = f_{i}, i = 1, 2.

Hence, β(u_{1}) ≤ β(u_{2}) and thus

(I − h\overline{L_{ϕ}}))^{−1}f_{1} ≤ (I − h\overline{L_{ϕ}}))^{−1}f_{2}.

Therefore, by induction,

(I − h\overline{L_{ϕ}}))^{−m}f_{1} ≤ (I − h\overline{L_{ϕ}}))^{−m}f_{2}.  (3.20)

Since L^{2}(H, μ) is dense in L^{1}(H, μ), (3.20) holds also for f_{1}, f_{2} ∈ L^{1}(H, μ). By taking f_{1}, f_{2} ∈ \overline{D(\overline{L_{ϕ}})}, we obtain as before that S_{t}(f_{1}) ≤ S_{t}(f_{2}). Proof of (iii). Arguing as before, it is sufficient to prove that

∫_{H}(I − h\overline{L_{ϕ}})^{−1}fdμ = ∫_{H} fdμ

for all h > 0 and f ∈ L^{2}(H, μ). This follows by integrating (3.17) over H to obtain

∫_{H} β(u)dμ = ∫_{H} fdμ,

hence our claim follows by integrating now (3.18) over H. □

4. A nonlinear eigenvalue problem associated with the Ornstein-Uhlenbeck operator

In this section we consider the nonlinear eigenvalue problem

−L_{2}u + β(u) = λu,  (4.1)

where β satisfies (H1) and is odd. By a solution to this equation we mean a pair (λ, u) ∈ ℝ × L^{2}(H, μ) satisfying u ∈ W^{2,2}(H, μ), β(u) ∈ L^{2}(H, μ). Clearly, for any λ ∈ ℝ, (λ, 0) is a solution to (4.1). Set

λ^{∗} := sup{λ ∈ ℝ : s ↦ β(s) − λs is strictly increasing in ℝ}

We have that 0 ≤ λ^{∗} < ∞. If λ < λ^{∗}, then s ↦ β(s) − λs is strictly increasing and hence, from Proposition 3.1 (a) we have that (λ, 0) is the only solution to (4.1). For λ ∈ ℝ let us consider the functional J_{λ} : L^{2}(H, μ) → [−∞, ∞] defined by

J_{λ}(u) = \begin{cases}J_{a}(u) + J_{b}(u) − \frac{λ}{2}∥u∥_{L^{2}(H,μ)}^{2}, & u ∈ W^{1,2}(H, μ), ∫_{H} b(u)dμ < ∞ \\ +∞ & otherwise. \end{cases}

(4.2)

We observe that for λ < λ^{∗}, J_{λ} is strictly convex, l.s.c. and nonnegative, and 0 is its global minimizer.

Next we investigate the positive constant solutions to (4.1) u(x) ≡ c. Then β(c) = λc. We have the following result.

Proposition 4.1. Assume that

t ↦ β(t)/t is strictly increasing on (0, ∞).  (4.3)

Then for all c > 0 the pair (β(c)/c, c) is a solution to (4.1) and u = c minimizes the functional J_{0} on the set

S_{c} := {u ∈ W^{1,2}(H, μ) : ∥u∥_{L^{2}(H,μ)} = c}.

Furthermore, u = c is the unique nonnegative minimizer of J_{0} on S_{c}.

Proof. From (4.3) we obtain that the mapping t ↦ b(\sqrt{t}), t > 0, is strictly convex, hence for any u ∈ D(J_{0}) we have by Jensen’s inequality ([10, Theorem 2.2(a)]) that

J_{0}(u) ≥ ∫_{H} b(\sqrt{|u|^{2}})dμ ≥ b(\sqrt{∫_{H} |u|^{2}dμ}) = b(c) = J_{0}(c),  (4.4)

implying that u = c is a minimizer for J_{0}. On the other hand, if u is a minimizer, then from (4.4) and the fact that J_{0}(c) ≥ J_{0}(u), we obtain that

∫_{H} b(\sqrt{|u|^{2}})dμ = b(\sqrt{∫_{H} |u|^{2}dμ}),

hence by [10, Theorem 2.2(b)] we deduce that u^{2} must be a constant, hence u = c since u is nonnegative. □

We will now state and prove our existence results.

Theorem 4.2. (i) For any R > 0 there exists a solution (λ, u) to (4.1) satis-

fying u ≥ 0 and u minimizes J_{0} on S_{R}. (ii) For any R > 0 there exists a sequence of solutions {(λ_{n}, u_{n})}_{n∈ℕ} to (4.1)

such that u_{n} ∈ S_{R} and
λ_{n} > 0 for n ∈ ℕ, and sup_{n∈ℕ} λ_{n} = ∞.  (4.5)

Proof. (ii) We will apply Theorem 1 in [4], see also [9]. As the real infinite dimensional separable Hilbert space we choose E = L^{2}(H, μ). Let φ : E → [0, ∞] be defined by φ(u) := J_{−1}, u ∈ E. Then clearly φ is convex, even, and φ(0) = 0. Moreover, in view of the compactness of the imbedding (2.5), the convex set

{u ∈ E : φ(u) ≤ ρ}

is compact in E for all ρ ≥ 0. Moreover, since E(H) ⊆ C_{b}(H) ∩ W^{1,2}(H, μ) we have that E(H) ⊆ D(φ). The density of E(H) in E implies the density of D(φ) in E.

Hence, all the assumptions of [4, Theorem 1] are satisfied and therefore there exists a sequence (ν_{k}, u_{k}) ∈ ℝ × E, k ∈ ℕ such that ∥u_{k}∥_{E} = R, ∂J_{−1}(u_{k}) ∋ ν_{k}u_{k} and sup_{k≥1} φ(u_{k}) = ∞. We claim that

D(∂J_{−1}) = D(L_{2}) ∩ D(B),

and

∂J_{−1}(u) = −L_{2}u + Bu + u, u ∈ D(∂J_{−1}).

Indeed,

J_{−1} = J_{a} + Jb̃,

where b̃(t) = b(t) + \frac{1}{2}t^{2} and we have

∂J_{a} = −L_{2}, and ∂Jb̃ = B + I.

In view of Proposition 3.1 (a), we have

R(−L_{2} + B + 2I) = E,

which implies that −L_{2} + (B + I) is maximal monotone. From [1, page 41] we have

∂J_{−1} = ∂J_{a} + ∂Jb̃,

which proves the claim. Therefore

−L_{2}u_{k} + β(u_{k}) = (ν_{k} − 1)u_{k}, k ∈ ℕ.

Set λ_{k} = ν_{k} − 1, k ∈ ℕ. By taking inner product with u_{k} and taking into account that ∥u_{k}∥_{E} = R > 0 we have that λ_{k} > 0. Finally, since

φ(u_{k}) ≤ ⟨∂φ(u_{k}), u_{k}⟩ = ν_{k}R^{2},

we have sup_{k∈ℕ} λ_{k} = ∞. and thus (4.5) holds.

(i) In this part we shall use that u ∈ W^{1,2}(H, μ) implies that |u| ∈ W^{1,2}(H, μ), J_{a}(|u|) = J_{a}(u), and moreover, since β is odd, we also have J_{b}(|u|) = J_{b}(u). We will apply Theorem 3 in [4]. To this end we set

P := \{v ∈ L^{2}(H, μ) : v ≥ 0\}, I_{P} (u) = \begin{cases}0 u ∈ P \\ +∞ otherwise, \end{cases}

and define φ_{+} : E → [0, ∞] by φ_{+}(u) = φ(u) + I_{P} (u), u ∈ E. We have that φ_{+} is convex, l.s.c., the set {u ∈ E : φ_{+}(u) ≤ ρ} is compact for every ρ ≥ 0, and φ_{+}(0) = 0. We claim that \overline{D(φ_{+})} = P. Indeed, let u ∈ P. By the density of D(φ) in E, there exists {u_{n}} ⊆ D(φ) such that u_{n} → u in E. Hence, |u_{n}| ∈ D(φ_{+}) and |u_{n}| → |u| = u in E.

Let R > 0. From [4, Theorem 3] there exists (ν, u) ∈ ℝ^{+} × P, with ∥u∥_{E} = R, νu ∈ D(∂φ_{+}), νu ∈ ∂φ_{+}(u) and

φ_{+}(u) = inf_{v∈S_{R}} φ_{+}(v).

It follows that

φ_{+}(v) ≥ φ_{+}(u) + ⟨νu, v − u⟩ for all v ∈ D(φ).

Since u ∈ P, we have φ_{+}(u) = φ(u), hence

φ_{+}(v) ≥ φ(u) + ⟨νu, v − u⟩ for all v ∈ D(φ).

Moreover, for all v ∈ P ∩ D(φ) we have

φ(v) ≥ φ(u) + ⟨νu, v − u⟩,

hence for all v ∈ D(φ) we have

φ(|v|) ≥ φ(u) + ⟨νu, |v| − u⟩.

Since φ(|v|) = φ(v), we obtain

φ(v) ≥ φ(u) + ⟨νu, v − u⟩ + ⟨νu, |v| − v⟩ ≥ φ(u) + ⟨νu, v − u⟩,

hence νu ∈ D(∂φ(u) and νu = −L_{2} + Bu + u. Setting now λ = ν − 1 we get

−L_{2}u + Bu = λu.

Finally, we have

J_{0}(u) = φ(u) − \frac{1}{2}R^{2} = φ_{+}(u) − \frac{1}{2}R^{2}

= inf_{v∈S_{R}} φ_{+}(v) − \frac{1}{2}R^{2} = inf_{v∈S_{R}} φ(|v|) − \frac{1}{2}R^{2}

= inf_{v∈S_{R}} φ(v) − \frac{1}{2}R^{2}

= inf_{v∈S_{R}} J_{0}(v).

□

We complete this note by exhibiting a class of functions β for which the minimum of J_{0} on S_{R} is not attained at the constants for R small.

Proposition 4.3. Assume that β satisfies the extra conditions

lim_{s→0} \frac{b(s)}{s^{2}} = ∞, and lim_{s→∞} \frac{b(s)}{s^{2}} = 0;  (4.6)
there exists C > 0 such that b(st) ≤ Cb(s)b(t) for all s, t > 0.  (4.7)

Then, there exists R_{0} > 0 such that for any R ∈ (0, R_{0}) J_{0} does not achieve its minimum on S_{R} at the constants.

Proof. For n ∈ ℕ, we set

ũ_{n}(t) = \begin{cases}−nα_{n}(|t| − \frac{1}{n}) & |t| ≤ \frac{1}{n} \\ 0 & |t| > \frac{1}{n} \end{cases}

where α_{n} will be chosen later. We define u_{n} : H → ℝ by u_{n}(x) := ũ_{n}(⟨x, e_{1}⟩) and we choose α_{n} so that ∥u_{n}∥_{L^{2}(H,μ)} = R. We observe also that u_{n} ∈ W^{1,2}(H, μ). One verifies that

C_{1}R\sqrt{n} ≤ α_{n} ≤ C_{2}R\sqrt{n},  (4.8)

for some positive constants C_{1}, C_{2}. We will show now that if n is chosen large enough and R > 0 is small enough, then

J_{0}(u_{n}) < J_{0}(R) = b(R).

Indeed, it follows from the definition of μ that

\frac{1}{2} ∫_{H} |Du_{n}|^{2}dμ ≤ K_{0} ∫_{0}^{1/n} |ũ′_{n}|^{2}dt, ∫_{H} b(u_{n})dμ ≤ K_{0} ∫_{0}^{1/n} b(ũ_{n})dt  (4.9)

for some positive constant K_{0}. Now, from (4.8) we have

∫_{0}^{1/n} |ũ′_{n}|^{2}dt = nα_{n}^{2} ≤ C_{2}^{2}n^{2}R^{2},  (4.10)

and from (4.8) and (4.7) we get

∫_{0}^{1/n} b(ũ_{n})dt = \frac{1}{nα_{n}} ∫_{0}^{α_{n}} b(s)ds ≤ \frac{b(α_{n})}{n} ≤ \frac{b(C_{2}R\sqrt{n})}{n} ≤ Cb(R)\frac{b(C_{2}\sqrt{n})}{n}.

Using now the second condition in (4.6) to find n_{0} ∈ ℕ so that

CK_{0} \frac{b(C_{2}\sqrt{n_{0}})}{n_{0}} < \frac{1}{4},

from the second inequality in (4.9) we obtain

∫_{H} b(u_{n})dμ ≤ ∫_{H} b(u_{n_{0}})dμ < \frac{1}{4}b(R).  (4.11)

Finally, in view of the first assumption in (4.6) we can find R_{0} > 0 such that for any R ∈ (0, R_{0})

K_{0}C_{2}^{2}n_{0}^{2} \frac{R^{2}}{b(R)} ≤ \frac{1}{4},

therefore from the first inequality in (4.9) and (4.10), we have

\frac{1}{2} ∫_{H} |Du_{n}|^{2}dμ ≤ K_{0}C_{2}^{2}n_{0}^{2} \frac{R^{2}}{b(R)}b(R) ≤ \frac{1}{4}b(R).  (4.12)

Hence, from (4.11) and (4.12) we conclude that for any R ∈ (0, R_{0}),

inf_{v∈S_{R}} J_{0}(v) ≤ J_{0}(u_{n_{0}}) ≤ \frac{1}{2}b(R) = \frac{1}{2}J_{0}(R).

This completes the proof of the proposition. □

Remark 4.4. We note that β(s) = |s|^{p−1}s, 0 < p < 1, satisfies all the assumptions of Proposition 4.3.

As a last comment, we mention that as a consequence of Theorem 4.2 and Proposition 4.3 we have shown the existence of a nonnegative nonconstant solution to (4.1). It is worth observing that a function u of the form

u(x) = ũ(⟨x, e_{1}⟩, ⟨x, e_{2}⟩, …, ⟨x, e_{N}⟩),

where ũ : ℝ^{N} → ℝ is a solution to (4.1) with H = ℝ^{N} with the usual inner product and

L_{2} = \frac{1}{2}Δ + ⟨b(x), ∇⟩,

where b_{i}(x) = −\frac{x_{i}}{2γ_{i}}, 1 ≤ i ≤ N, is also a solution to the infinite dimensional problem. It is an open problem to know whether (4.1) possesses solutions depending on infinitely many variables.

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Philippe Clément Mathematical Institute, Leiden University, P.O. Box 9512, NL-2300, RA Leiden, and EEMCS/ DIAM, TU Delft, P.O. Box 5031, NL-2600, GA Delft, The Netherlands

E-mail address: clement@math.leidenuniv.nl

Marta García-Huidobro Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

E-mail address: mgarcia@mat.puc.cl

Raúl F. Manásevich Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile

E-mail address: manasevi@dim.uchile.cl