Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 201–221 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Existence and number of solutions to semilinear equations with applications to boundary-value

problems ^{∗}

P. S. Milojević

Abstract

We present recent and some new existence results on the number of solutions to nonlinear equations and to (non)resonant semilinear equations involving nonlinear perturbations of Fredholm maps of index zero. We apply our results to semilinear elliptic, and to semilinear parabolic and hyperbolic periodic boundary-value problems.

1 Introduction

Let X and Y be Banach spaces and T : X → Y be a nonlinear map of A-proper type. Under various conditions on T, we study in Section 2 the surjectivity and the finitness of the solution set of the equation T x = f. In particular, we look at nonresonant semilinear equations of the form Ax + Nx = f where A is a Fredholm map of index zero and the nonlinear map N is such that A + N is (pseudo) A-proper. We say that this equation is not at resonance if A and N are are such that it is solvable for each f ∈ Y. Applications to semiabstract nonresonant semilinear equations are given in Section 3. Section 4 is devoted to applications of the results of Section 3 to boundary-value problems (BVP) for semilinear elliptic equations. In Section 5, some comments on periodic BVP’s for semilinear parabolic and hyperbolic equations assuming nonuniform nonresonance conditions are made. The existence of solutions for such problems has been studied earlier in [12, 13, 14, 22, 6, 7, 9].

2 Number of solutions to operator equations

In this section, we shall study the number of solutions to the equation T x = f. The unique (approximate) solvability of this equation has been studied in detail in [20], using the A-proper mapping approach.

∗1991 Mathematics Subject Classifications: 47H15, 47H09, 35J40.

Key words: Existence, number of solutions, (pseudo) A-proper maps, elliptic BVP’s,

parabolic and hyperbolic equations.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University.

Published October 25, 2000.

Definition A map T : D ⊂ X → Y is (pseudo) A-proper with respect to a scheme Γ = {X_{n}, Y_{n}, Q_{n}} with dim X_{n} = dim Y_{n} on D if whenever {x_{n_{k}} ∈ D ∩ X_{n_{k}}} is bounded and such that Q_{n_{k}}T x_{n_{k}} − Q_{n_{k}}f → 0 for some f ∈ Y, then {x_{n}} has a subsequence converging to x ∈ D (there is x ∈ D) with T x = f.

Next, we shall define A-proper homotopies.

Definition A homotopy H : [0, 1] × D → Y is A-proper with respect to Γ on D if Q_{n}H_{t} : D∩X_{n} → Y_{n} is continuous for each t and n, and if {x_{n_{k}} ∈ D∩X_{n_{k}}} is bounded and t_{k} ∈ [0, 1] with t_{k} → t are such that Q_{n_{k}}H(t_{k}, x_{n_{k}}) − Q_{n_{k}}f → 0 as k → ∞ for some f ∈ Y, then a subsequence of {x_{n_{k}}} converges to x ∈ D and H(t, x) = f.

The classes of A-proper and pseudo A-proper maps are very general. For many examples of such maps, we refer the reader to [15]-[19].

Nonlinear equations

We say that a map T : X → Y satisfies condition (+) if {x_{n}} is bounded whenever T x_{n} → f in Y. Let Σ be the set of all points x ∈ X where T is not locally invertible and cardT^{−1}({f}) be the cardinal number of the set T^{−1}({f}).

Theorem 2.1 ([21]) Let T : X → Y be continuous, A-proper and satisfy condition (+). Then

(a) The set T^{−1}({f}) is compact (possibly empty ) for each f ∈ Y.

(b) The range R(T) of T is closed and connected.

(c) Σ and T (Σ) are closed subsets of X and Y, respectively, and T (X \ Σ) is open in Y.

(d) cardT^{−1}({f}) is constant and finite (it may be 0) on each connected component of the open set Y \ T (Σ).

(e) if Σ = ∅, then T is a homeomorphism from X to Y.

(f) if Σ ≠ ∅, then the boundary ∂T (X \ Σ) of T (X \ Σ) satisfies ∂T (X \ Σ) ⊂ T (Σ).

Proof. Since T is proper by Proposition 2.1 in [21], it is a closed map. Since X \ Σ is an open set, Σ is a closed set. Hence (a)-(c) hold, where T (X \ Σ) is open since T is locally invertible on X \ Σ. (d) follows from the Ambrosetti theorem [A] and (e) follows from the global inversion theorem. Next, (b) and (c) imply that

T (X) = T (Σ) ∪ T (X \ Σ) = T (Σ) ∪ \overline{F (X \ Σ)} = \overline{T (X)}.  (2.1)

Moreover, ∂T (X \ Σ) = \overline{T (X \ Σ)} \ T (X \ Σ), which together with (2.1) imply (f). ♢

Next, we shall look at another surjectivity result. Let J : X → 2^{X^{∗}} be the normalized duality map and G : X → Y be a bounded map such that Gx ≠ 0 for all x with ∥x∥ ≥ r_{0} for some r_{0} > 0 and

For each large r > 0, deg(Q_{n}G, B(0, r) ∩ X_{n}, 0) ≠ 0 for all large n. (2.2)

Theorem 2.2 Let T : X → Y satisfy conditions (+) and (2.2), and let

(i) For each f ∈ Y there is an r_{f} > 0 such that

T x ≠ λGx for x ∈ ∂B(0, r_{f}), λ < 0.  (2.3)

(ii) H(t, x) = tT x + (1 − t)Gx is an A-proper with respect to Γ homotopy on [0, 1] × X.

Then T is surjective. Moreover, if T is continuous, then T^{−1}({f}) is compact for each f ∈ Y and the cardinal number cardT^{−1}({f}) is constant, finite and positive on each connected component of the set Y \ T (Σ).

Proof. The surjectivity of T has been established earlier by the author (see, eg [17, 19]). Moreover, T is continuous and proper by Proposition 2.1 in [21]. Hence, the other assertions of the theorem follow from Theorem 2.1.

Corollary 2.1 Let F, K : X → X be continuous ball-condensing maps and T = I − F and G = I − K satisfy (2.2)-(2.3). Then the conclusions of Theorem 2.2 hold for T.

This corollary is also valid for general condensing maps (see [23]). For a map M, define its quasinorm by |M| = lim sup_{∥x∥→∞} ∥Mx∥/∥x∥.

Theorem 2.3 (cf. [19]) Let A : D(A) ⊂ X → Y be a linear densely defined map and N : X → Y be bounded and of the form Nx = B(x)x + Mx for some linear maps B(x) : X → X. Assume that there is a c > |M| and a positively homogeneous map C : X → Y such that

∥Ax − (1 − t)Cx − tB(x)x∥ ≥ c∥x∥, x ∈ D(A) \ B(0, R).  (2.4)

(i) H_{t} = A − (1 − t)C − tN is A-proper with respect to Γ = {X_{n}, Y_{n}, Q_{n}} for t ∈ [0, 1) and A − N is pseudo A-proper

(ii) For all r > R, deg(Q_{n}(A − C), B(0, r) ∩ X_{n}, 0) ≠ 0 for each large n.

Then the equation Ax − Nx = f is solvable for each f ∈ Y. If, in addition, A − N is continuous and A-proper, then (A − N)^{−1}({f}) is compact for each f ∈ Y and card(A−N)^{−1}({f}) is constant, finite and positive on each connected component of the set Y \ (A − N)(Σ).

Proof. Regarding the surjectivity of A − N, it suffices to solve Ax − Nx = 0. Define H(t, x) = Ax − (1 − t)Cx − tNx on [0, 1] × D(A). Then there is an r > 0 such that

H(t, x) ≠ 0 for x ∈ ∂B(0, r) ∩ D(A), t ∈ [0, 1].  (2.5)

If not, then there are x_{n} ∈ H and t_{n} ∈ [0, 1] such that ∥x_{n}∥ → ∞ and H(t_{n}, x_{n}) = 0. Let ϵ > 0 be small such that |M| ≤ (|M| + ϵ)∥x∥ for ∥x∥ ≥ R_{1} and |M| + ϵ < c. For each x_{n} with ∥x_{n}∥ ≥ R_{1} we have that

c∥x_{n}∥ ≤ ∥Ax_{n} − (1 − t)Cx_{n} − tB(x_{n})x_{n}∥ ≤ (|M| + ϵ)∥x_{n}∥.

Dividing by ∥x_{n}∥, this leads to a contradiction and (2.5) holds. Hence, A − N is surjective by the homotopy result in [16, 17]. Next, it is easy to see that ∥(A − N)x∥ → ∞ as ∥x∥ → ∞ by (2.4). Hence, the other assertions follow from Theorem 2.1.

3 Semi-abstract nonresonance problems

Let Q ⊂ R^{n} be a bounded domain, V be a closed subspace of W_{2}^{2m}(Q) containing the test functions and L : V → L_{2} be a linear map with closed range in H = L_{2}(Q). Let V_{1} be a closed subspace of V and L_{1} be the restriction of L to V_{1}. Assume

(L1) Each eigenvalue λ_{j} of L_{1} has a finite multiplicity and the corresponding eigenfunctions {…, w_{−1}, w_{0}, w_{1}, …} form a complete set in V_{1}.

Let A = A_{1} + L for some linear map A_{1} : V → H. For a fixed integer j, define B : V → H by Bu = −Au + λ_{j}u.

(B1) There is λ ≠ λ_{j}, j = 1, 2, …, such that the map B − λI = −A_{1} − L − (λ − λ_{j})I : V → L_{2} is bijective.

Let λ ≠ λ_{j} for each j = 1, 2, … be fixed, Γ = {Y_{n}, Q_{n}} be a projectionally complete scheme for L_{2} and X_{n} = (B − λI)^{−1}(Y_{n}) ⊂ V for each n. Then Γ_{B} = {X_{n}, Y_{n}, Q_{n}} is an admissible or a projectionally complete scheme for (V, L_{2}). Since B − λI : V → L_{2} is linear, one-to-one and A-proper with respect to Γ_{B}, there is a constant c > 0 (depending)only on λ) such that

∥(B − λI)u∥ ≥ c∥u∥_{V} , u ∈ V.  (3.1)

Consider the following semilinear equation in V

Au + g(x, u, Du, …, D^{2m−1}u)u + f(x, u, Du, …, D^{2m}u) = h(x)  (3.2)

For u ∈ H, set u^{±} = max(±u, 0). Let r = λ_{j+1} − λ_{j}. We require that B has the following properties:

Property I B is a closed densely defined map in H with closed range R(B), (Bu, u) ≥ −r^{−1}∥Bu∥^{2} and R(B) = N(B)^{⊥} in H, N(−L_{1} + λ_{j}I) ⊂ N(B) and (Bu, u) = (−Lu + λ_{j}u, u) on V.

Property II If (Bu, u) = −r^{−1}∥Bu∥^{2} for some u ∈ V, then u ∈ N(−L_{1} + λ_{j}I) ⊕ N(−L_{1} + λ_{j+1}I).

Let us note that if B^{−1} is a partial inverse of B and B^{−1} + r^{−1}I is strongly monotone on R(B), i.e. it is a bounded linear map on R(B) and ((B^{−1} + r^{−1}I)u, u) = c_{0}∥(B^{−1}+r^{−1}I)u∥^{2} on R(B) for some c_{0} > 0, then ([BF]) Property II holds in the sense that if (Bu, u) = −r^{−1}∥Bu∥^{2} for some u ∈ V, then u ∈ N(B) ⊕ N(B + rI). If B is selfadjoint or angle bounded in the sense of H. Amann, it is known that B^{−1} + r^{−1}I is strongly monotone. If B ≠ B^{∗} and B is a normal map, the strong monotonicity of B^{−1} + r^{−1}I has been discussed in Hetzer [8].

Some properties of B are given next.

Lemma 3.1 Let B have Properties I and II. Suppose that p_{±} ∈ L_{∞}(Q) are such that 0 ≤ p_{±}(x) ≤ r for a.e. x ∈ Q and

∫_{Q}[p_{+}(v^{+})^{2} + p_{−}(v^{−})^{2}] > 0 for all v ∈ N(−L_{1} + λ_{j}I) \ {0}

and

∫_{Q}[(r − p_{+})(w^{+})^{2} + (r − p_{−})(w^{−})^{2}] > 0 for all w ∈ N(−L_{1} + λ_{j+1}I) \ {0}.

Then the equation

Bu + p_{+}u^{+} − p_{−}u^{−} = 0  (3.3)

has only the trivial solution.

Proof. Define p : Q × R → R by

p(x, u) = p_{+}(x) if u ≥ 0,
p(x, u) = p_{−}(x) if u ≤ 0.

Then

0 ≤ p(x, u) ≤ r for (x, u) ∈ Q × R  (3.4)

and, for u ∈ H and a.e. x ∈ Q,

p(x, u(x))u(x) = p(x, u(x))u^{+}(x) − p(x, u(x))u^{−}(x)
= p_{+}(x)u^{+}(x) − p_{−}(x)u^{−}(x).

Define P : V ⊂ H → H by (P u)(x) = p(x, u(x))u(x) for a.e. x ∈ Q. Then (3.3) is equivalent to

Bu + P u = 0, u ∈ V.  (3.5)

By (3.4), we have that ∥P u∥^{2} ≤ r(P u, u) on V. Moreover, for each solution u ∈ V of (3.5), we get by Property I that

−r^{−1}∥P u∥^{2} = −r^{−1}∥Bu∥^{2} ≤ (Bu, u) = (−P u, u)

and so ∥P u∥^{2} ≥ r(P u, u). Hence, ∥P u∥^{2} = r(P u, u) and (Bu, u) = −r^{−1}∥Bu∥^{2}. By Property II, we get that u ∈ N(−L_{1} + λ_{j}I) ⊕ N(−L_{1} + λ_{j+1}I). Hence, u = v + w with v ∈ N(−L_{1} + λ_{j}I) and w ∈ N(−L_{1} + λ_{j+1}I). Since u is a solution of (3.3), we get that

(Bu, u) = (−Lu + λ_{j}u, u)
= (−Lv + λ_{j}v, v) + (−Lv + λ_{j}v, w) + (−Lw + λ_{j}w, v + w)
= (−Lw + λ_{j+1}w − rw, v + w)
= (−rw, w)

and so (−rw, w) + (p(., u(.))(v + w), v + w) = 0. Then

(v − w, −rw + p(., u(.))(v + w))
= (v + w, −rw + p(., u(.))(v + w)) − 2(w, −rw + p(., u(.))(v + w))
= −2(w, −rw + p(., u(.))(v + w))
= −2(v + w, −rw − B(v + w)) + 2(v, −rw − B(v + w))
= −2(v, rw + B(v + w)) = −2(v, B(v + w)) = 0

since v ∈ N(−L_{1} + λ_{j}I) ⊂ N(B) and R(B) = N(B)^{⊥}. Since

(p(., u(.))(v + w), v − w) = (p(., u(.))v, v) + ([r − p(., u(.))]w, w)
+([r − p(., u(.))]w, −v) + (p(., u(.))v, −w)
= (p(., u(.))v, v) + ([r − p(., u(.))]w, w)

we get that

(p(., u(.))v, v)+([r−p(., u(.))]w, w) = (p(., u(.))(v+w), v−w)+(rw, −v+w) = 0.

(3.6)

Since each term in (3.6) is nonnegative by (3.4), we get that each term is zero, i.e.,

∫_{Q} p(x, v(x) + w(x))v^{2}(x)dx = 0  (3.7)

∫_{Q}[(r − p(x, v(x) + w(x))]w^{2}(x)dx = 0.  (3.8)

Set Q_{v} = {x ∈ Q |v(x) ≠ 0} and Q_{w} = {x ∈ Q |w(x) ≠ 0}.

By (3.7)-(3.8), we get p(x, v(x) + w(x)) = 0 for a.e. x ∈ Q_{v} and p(x, v(x) + w(x)) = r for a.e. x ∈ Q_{w} and so Q_{v} ∩ Q_{w} = ∅. If Q_{v} = ∅, then u = w and the equation (3.8) becomes

0 = ∫_{Q}[(r − p(x, w(x))]w^{2}(x)dx = ∫_{Q}(r − p_{+})(w^{+})^{2} + (r − p_{−})(w^{−})^{2}

so that by our hypothesis, w = 0 and therefore u = 0.

Next, suppose that Q_{v} ≠ ∅. Then we have that p(x, v(x) + w(x)) = 0 on Q_{v} and, by (3.8), ∫_{Q_{v}} rw^{2}(x) = 0, i.e., w(x) = 0 for a.e. x ∈ Q_{v}. Then by (3.7)

0 = ∫_{Q_{v}} p(x, v(x))v^{2}(x) = ∫_{Q_{v}}(p_{+}(v^{+})^{2} + p_{−}(v^{−})^{2}) = ∫_{Q}(p_{+}(v^{+})^{2} + p_{−}(v^{−})^{2}).

By our assumption, this implies that v = 0, in contradiction to Q_{v} ≠ ∅. Hence, Q_{v} = ∅ and u = 0.

Lemma 3.2 Let (L1) and (B1) hold and B have Properties I and II. Suppose that a_{±}, b_{±} ∈ L_{∞}(Q) are such that 0 ≤ a_{±}(x) ≤ b_{±} ≤ r for a.e. x ∈ Q and

∫_{Q}[a_{+}(v^{+})^{2} + a_{−}(v^{−})^{2}] > 0 for all v ∈ N(−L_{1} + λ_{j}I) \ {0}  (3.9)

and

∫_{Q}[(r−b_{+})(w^{+})^{2}+(r−b_{−})(w^{−})^{2}] > 0 for all w ∈ N(−L_{1}+λ_{j+1}I)\{0}. (3.10)

Then there exists ϵ = ϵ(a_{±}, b_{±}) > 0 and δ = δ(a_{±}, b_{±}) > 0 such that for all p_{±} ∈ L_{∞}(Q) with

a_{+}(x) − ϵ ≤ p_{+}(x) ≤ b_{+}(x) + ϵ (3.11)
a_{−}(x) − ϵ ≤ p_{−}(x) ≤ b_{−}(x) + ϵ (3.12)

for a.e. x ∈ Q and for all u ∈ V, one has

∥Bu + p_{+}u^{+} − p_{−}u^{−}∥ ≥ δ∥u∥_{V} .  (3.13)

Proof. If this is not the case, then we can find the sequences {u_{k}} ⊂ V, with ∥u_{k}∥_{V} = 1 for each k and {p_{±}^{k}} ⊂ L_{∞}(Q) such that

a_{±}(x) − k^{−1} ≤ p_{±}(x) ≤ b_{±}(x) + k^{−1} a.e. on Q  (3.14)

and

Bu_{k} + p_{+}^{k}u_{k}^{−} − p_{−}^{k}u_{k}^{−} = v_{k} → 0 as k → ∞.  (3.15)

Then p_{±}^{k} → p_{±} weakly in H with a_{±}(x) ≤ p_{±}(x) ≤ b_{±}(x) a.e. on Q. Let μ ≠ λ_{j} and consider the identity

u_{k} + (B − μI)^{−1}[(p_{+}^{k} − p_{+})u_{k}^{+} − (p_{−}^{k} − p_{−})u_{k}^{−}] (3.16)

= (B − μI)^{−1}(−p_{+}u_{k}^{+} + p_{−}u_{k}^{−} − μu_{k} + v_{k}).

By the compactness of the embedding of V into L_{2}, we have that u_{k} → u in L_{2} as well as u_{k}^{±} → u^{±} in L_{2}. Since (B − μI)^{−1} is continuous both as a map from L_{2} to V and from L_{2} to L_{2}, we get that

(B − μI)^{−1}(−p_{+}u_{k}^{+} + p_{−}u_{k}^{−} − μu_{k} + v_{k}) → (B − μI)^{−1}(−p_{+}u^{+} + p_{−}u^{−} − μu) (3.17) in L_{2} and V. Next, we shall show that p_{±}^{k} → p_{±}u^{±} weakly in H. For ϕ ∈ C_{0}^{∞}(Q), we have that

(p_{+}^{k}u_{k}^{+} − p_{+}u^{+}, ϕ) = (p_{+}^{k}(u_{k}^{+} − u^{+}), ϕ) + ((p_{+}^{k} − p_{+})u^{+}, ϕ)
≤ c∥u_{k}^{+} − u^{+}∥ + ((p_{+}^{k} − p_{+})u^{+}, ϕ)

which approaches zero as k approaches ∞. Hence, p_{+}^{k}u_{k}^{+} → p_{+}u^{+} weakly in L_{2} by the density of C_{0}^{∞}(Q) in L_{2}, and similarly, p_{−}^{k}u_{k}^{−} → p_{−}u^{−} weakly in L_{2}. Hence, (3.16)-(3.17) imply that u = (B − μI)^{−1}(−p_{+}u^{+} + p_{−}u^{−} − μu), i.e., Bu + p_{+}u^{+} − p_{−}u^{−} = 0. Moreover, for each v ∈ N(−L_{1} + λ_{j}I) \ {0}, we have that

∫_{Q} p_{+}(v^{+})^{2} + p_{−}(v^{−})^{2} ≥ ∫_{Q} a_{+}(v^{+})^{2} + a_{−}(v^{−})^{2} > 0

and, for each w ∈ N(−L_{1} + λ_{j+1}I) \ {0}, we have that

∫_{Q}[(r − p_{+})(w^{+})^{2} + (r − p_{−})(w^{−})^{2}] ≥ ∫_{Q}[(r − b_{+})(w^{+})^{2} + (r − b_{−})(w^{−})^{2}] > 0.

Hence, by Lemma 3.1, u = 0 a.e.on Q. Thus, u_{k} → 0 in L_{2}, ∥u_{k}∥_{V} = 1 and ∥p_{+}^{k}u_{k}^{+} − p_{−}^{k}u_{k}^{−} − μu_{k}∥ → 0. By (3.1), we get that

∥Bu_{k} + p_{+}^{k} − p_{−}^{k}u_{k}^{−}∥ ≥ ∥Bu_{k} − μu_{k}∥ − ∥μu_{k} − p_{+}^{k}u_{k}^{+} + p_{−}^{k}u_{k}^{−}∥
≥ c − ∥p_{+}^{k}u_{k}^{+} − p_{−}^{k}u_{k}^{−} − μu_{k}∥.

By (3.15), passing to the limit as k → ∞, we get that 0 ≥ c > 0, a contradiction. Hence, the lemma is valid.

Remark 3.1 Modifying suitably the proof of Lemma 3.2, condition (B1) can be replaced by

(B2) dim N(B) < ∞ and the partial inverse of B is compact.

Let R^{s_{k}} be the vector space whose elements are ξ = {ξ_{α} : |α = (α_{1}, …, α_{n})| ≤ k}. Each ξ ∈ ℝ^{k} may be written as a pair ξ = (η, ζ) with η ∈ R^{s_{k−1}}, ζ = {ξ_{α} | |α| = k} ∈ R^{s_{k}−s_{k−1}} = R^{s′_{k}} and |ξ| = (∑_{|α|≤k} |ξ_{α}|^{2})^{1/2}. Set η(u) = (Du, …, D^{2m−1}u) and ξ(u) = (u, Du, …, D^{2m}u). Define Nu = g(x, u, η(u))u+ F u, where F u = f(x, ξ(u)). Set k = s_{2m−1} − 1.

For our next result, suppose that dim N(−L_{1} + λ_{j}I) = 1 and is spanned by a positive function w_{j}.

Lemma 3.3 Let B have properties I and II. Suppose that p_{±} ∈ L_{∞}(Q) are such that 0 ≤ p_{±}(x) ≤ r a.e. x ∈ Q and for r = λ_{j+1} − λ_{j}

∫ [(r − p_{+})(w^{+})^{2} + (r − p_{−})(w^{−})^{2}] > 0 for all w ∈ N(−L_{1} + λ_{j+1}) \ {0}.

Then, if u is a solution of Bu + p_{+}u^{+} − p_{−}u^{−} = 0, then u ∈ N(−L_{1} + λ_{j}I).

Proof. If not, then arguing as in Lemma 3.1 we get that u = 0 if Q_{v} = ∅. Next, suppose that Q_{v} ≠ ∅. Then it follows from the properties of the eigenfunction w_{j} and the fact that v = aw_{j}(x) for some a ∈ R and therefore Q_{v} = Q. Hence, we must have that p(x, u(x) = 0 for a.e. x ∈ Q. By (3.8), we get that w = 0 and hence u = v. Thus, in both cases, u = v ∈ N(−L_{1} + λ_{j}I).

Theorem 3.1 Let (L1) and (B1) hold, B have properties I and II and there are functions γ_{±}, Γ_{±} ∈ L_{∞}(Q) such that for some j ∈ J ⊂ Z, one has

λ_{j} ≤ γ_{±}(x) ≤ Γ_{±}(x) ≤ λ_{j+1} for a.e. x ∈ Q ,

∫_{Q}[(γ_{+} − λ_{j})(v^{+})^{2} + (γ_{−} − λ_{j})(v^{−})^{2}] > 0 (3.18)

for all v ∈ N(L_{1} − λ_{j}I) \ {0}, and

∫_{Q}[(λ_{j+1} − Γ_{+})(w^{+})^{2} + (λ_{j+1} − Γ_{−})(w^{−})^{2}] > 0  (3.19)

for all w ∈ N(L_{1} − λ_{j+1}I) \ {0}. Also suppose that for ϵ > 0 and δ > 0 given in Lemma 3.2,

(G1) there is ρ > 0 such that for a.e. x ∈ Q

γ_{+}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{+}(x) + ϵ if u > ρ, η(u) ∈ ℝ^{k}
γ_{−}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{−}(x) + ϵ if u < −ρ, η(u) ∈ ℝ^{k}

(G2) There are functions b(x) ∈ L_{∞}(Q) and k_{s}(x) ∈ L_{2}(Q) for each s > 0 such that

|g(x, u, η(u))| ≤ sb(x)( ∑_{|α|≤2m−1} |D^{α}u|^{2})^{1/2} + k_{s}(x), u ∈ V.

(F) ∥F u∥ = ∥f(x, u, Du, …, D^{2m}u)∥ ≤ β∥u∥_{V} + γ for β ∈ (0, δ), γ > 0.

(H) H_{t} = A − λ_{j}I − tF : V → H is A-proper with respect to Γ_{B} for t ∈ [0, 1) and H_{1} = B − F is pseudo A-proper.

Then (3.2) has at least one solution in V for each h ∈ H. If H_{1} is A-proper, the set of solutions S(h) of (3.2) is compact for each h ∈ L_{2} and card S(h) is constant, finite and positive on each connected component of the set L_{2} \ (A − N)(Σ).

Proof. Let g_{1} : Q×R×ℝ^{k} → R be given by g_{1}(x, u, η(u)) = g(x, u, η(u))−λ_{j}. Then define functions

g_{+}(x, u, η(u)) = g_{1}(x, u, η(u)) for all (x, η(u)) ∈ Q × ℝ^{k}, u ≥ ρ

g_{+}(x, u, η(u)) = g_{1}(x, ρ, η(u)) for all (x, η(u)) ∈ Q × R^{s_{2m−1}}, 0 ≤ u ≤ ρ ,

g_{−}(x, u, η(u)) = g_{1}(x, u, η(u)) for all (x, η(u)) ∈ Q × ℝ^{k}, u ≤ −ρ , g_{−}(x, u, η(u)) = −g_{1}(x, −ρ, η(u)) for all (x, η(u)) ∈ Q × R^{s_{2m−1}}, −ρ ≤ u ≤ 0 ,

q(x, 0, η(u)) = g_{1}(x, 0, η(u)) for all (x, η(u)) ∈ Q × ℝ^{k} , q(x, u, η(u)) = g_{1}(x, u, η(u))u − g_{+}(x, u, η(u))u for all (x, η(u)) ∈ Q × ℝ^{k} and u > 0. Also define

q(x, u, η(u)) = g_{1}(x, u, η(u))u − g_{−}(x, u, η(u))u for all (x, η(u)) ∈ Q × ℝ^{k}

and u < 0. Then q satisfies Caratheodory conditions. Set a_{±}(x) = γ_{±}(x) − λ_{j} and b_{±}(x) = Γ_{±}(x) − λ_{j}. Then

a_{+}(x) − ϵ ≤ g_{+}(x, u, η(u)) ≤ b_{+}(x) + ϵ on Q × R_{+} × ℝ^{k}
a_{−}(x) − ϵ ≤ g_{−}(x, u, η(u)) ≤ b_{−}(x) + ϵ on Q × R_{−} × ℝ^{k}.

Then in V, problem (3.2) is equivalent to

Bu + g_{+}(x, u^{+}, η(u))u^{+} − g_{−}(x, −u^{−}, η(u))u^{−} + f(x, ξ(u)) + q(x, u, η(u)) = −h.

Then for u ∈ H, set Q_{+}(u) = {x ∈ Q | u(x) > 0}, Q_{−}(u) = {x ∈ Q | u(x) < 0} and let χ_{Q_{±}} be the corresponding characteristic functions. Define the maps E : H → L_{∞}(Q), F, G, H : V → H, respectively, by

E(u)(x) = g_{+}(x, u^{+}(x), η(u))χ_{Q_{+}(u)} + g_{−}(x, −u^{−}(x), η(u))χ_{Q_{−}(u)}

G(u)(x) = [E(u)(x)]u(x) = (E(u)u)(x) so that

G(u)(x) = g_{+}(x, u^{+}(x), η(u))u^{+}(x) − g_{−}(x, −u^{−}(x), η(u))u^{−}(x),

F (u)(x) = f(x, ξ(u)) and H(u)(x) = q(x, u(x), η(u)). Hence (3.2) can be written in the operator form

Bu + Gu + F u + Hu = −h, u ∈ V.  (3.20)

We know that G, F and H are well defined, continuous and bounded in H.

Let C : H → H be defined by C(u)(x) = b_{+}(x)u^{+}(x)−b_{−}(x)u^{−}(x). Clearly, C is a positively homogeneous map and C, G, H : V → H are completely continuous maps, i.e. they map weakly convergent sequences in V into strongly convergent sequences in H. Indeed, let us show this, for example, for G. Since V is compactly embedded in H, it follows from the construction of G and (G2) that if {u_{k}} ⊂ V converges weakly to u_{0} in V, then ([K])

∥g_{+}(x, u_{k}^{+}, η(u_{k})) − g_{−}(x, −u_{k}^{−}, η(u_{k})) − g_{+}(x, u_{0}^{+}, η(u_{0})) + g_{−}(x, −u_{0}^{−}, η(u_{0}))∥

approaches 0. Hence, the map G : V → L_{p} is completely continuous since

∥Gu_{k} − Gu_{0}∥ = ∥E(u_{k})u_{k} − E(u_{0})u_{0}∥
≤ ∥E(u_{k})(u_{k} − u_{0})∥ + ∥E(u_{k}) − E(u_{0})∥∥u_{0}∥
≤ max{∥a_{+}∥_{∞} + ∥a_{−}∥_{∞} + 2ϵ, ∥b_{+}∥_{∞} + ∥b_{−}∥_{∞}
+2ϵ}∥u_{k} − u_{0}∥ + ∥E(u_{k}) − E(u_{0})∥ ∥u_{0}∥ → 0 .

Thus, we have that H_{t} = B + (1 − t)C + t(F + G + H) is A-proper for each t ∈ [0, 1) from V → H and H_{1} : V → H is pseudo A-proper.

Next, by construction

(1 − t)Cu + tGu = [(1 − t)b_{+}(x) + tg_{+}(x, u^{+}, η(u))]u^{+}(x)
−[(1 − t)b_{−}(x) + tg_{−}(x, −u^{−}, η(u))]u^{−}(x)

and, for a.e. x ∈ Q, η(u) ∈ ℝ^{k}

a_{+}(x) − ϵ ≤ (1 − t)b_{+}(x) + tg_{+}(x, u^{+}(x), η(u)) ≤ b_{+}(x) + ϵ
a_{−}(x) − ϵ ≤ (1 − t)b_{−}(x) + tg_{−}(x, −u^{−}(x), η(u)) ≤ b_{(}x) + ϵ

and |q(x, u, η(u))| ≤ d_{ρ}(x) for a.e. x ∈ Q and all (u, η(u)) ∈ R × ℝ^{k}, where d_{ρ} ∈ L_{2}(Q) is independent of u since g (and hence g_{1}) grows at most linearly. Hence, by Lemma 3.2 with p_{+}(x) = (1 − t)b_{+}(x) + tg_{+}(x, u^{+}(x), η(u)) and p_{−}(x) = (1 − t)b_{−}(x) + tg_{−}(x, −u^{−}(x), η(u)), we get for some c > 0

∥Bu + (1 − t)Cu + tGu∥ ≥ δ∥u∥^{2} for all u ∈ V.

It is left to show that deg(Q_{n}(B +C), B_{R} ∩V_{n}, 0) ≠ 0 for all n. Let η ∈ (0, r) be fixed. Then, for each t ∈ [0, 1], and a.e. x ∈ Q, we have that 0 ≤ (1−t)η+tb_{±} ≤ r. It is easy to show that p_{+} = (1 − t)η + tb_{+} and p_{−} = (1 − t)η + tb_{−} satisfy 0 ≤ p_{±} ≤ r for a.e. x ∈ Q, and

∫_{Q}[p_{+}(v^{+})^{2} + p_{−}(v^{−})^{2}] > 0 for all v ∈ N(−L_{1} + λ_{j}I) \ {0}

and

∫_{Q}[r − p_{+})(w^{+})^{2} + (r − p_{−})(w^{−})^{2}] > 0 for all w ∈ N(−L_{1} + λ_{j+1}I) \ {0}.

Hence, one gets that the equation

Bu + [(1 − t)η + tb_{+}]u^{+} + [(1 − t)η + tb_{−}]u^{−} = 0  (3.21)

has only the trivial solution for each t ∈ [0, 1]. Since the homotopy given by (3.21) is A-proper, there is an n ≥ n_{0} such that for each R > 0 and all n ≥ n_{0},

deg(P_{n}(B + b_{+}(.)^{+} − b_{−}(.)^{−}, B(0, R) ∩ H_{n}, 0)
= deg(P_{n}(B + ηI), B(0, R) ∩ H_{n}, 0) = ±1 .

Hence, (3.2) is solvable in V by Theorem 2.3. The other assertions also follow from this theorem.

Remark 3.2 Conditions (3.18)-(3.19) hold for a wide class of nonlinearities g. For example, they are implied by λ_{j} < λ_{j} +ϵ ≤ γ_{+}(x) ≤ Γ_{+}(x) ≤ λ_{j+1} and λ_{j} ≤ γ_{−}(x) ≤ Γ_{−}(x) ≤ λ_{j+1} − ϵ < λ_{j+1}, or λ_{j} ≤ γ_{+}(x) ≤ Γ_{+}(x) ≤ λ_{j+1} − ϵ < λ_{j+1} and λ_{j} < λ_{j} + ϵ ≤ γ_{−}(x) ≤ Γ_{−}(x) ≤ λ_{j+1}, in the case when the eigenfunctions associated to λ_{j} and λ_{j+1} change sign in Q.

Next, we shall give some concrete assumptions on f and g that imply (F)-(H) in Theorem 3.1.

Theorem 3.2 Assume that (L1) and (B1) hold, B have properties I and II and there be functions γ_{±}, Γ_{±} ∈ L_{∞}(Q) such that for some j ∈ J ⊂ Z, one has

λ_{j} ≤ γ_{±}(x) ≤ Γ_{±}(x) ≤ λ_{j+1} for a.e. x ∈ Q

∫_{Q}[(γ_{+} − λ_{j})(v^{+})^{2} + (γ_{−} − λ_{j})(v^{−})^{2}] > 0 for all v ∈ N(L_{1} − λ_{j}I) \ {0} ,

∫_{Q}[(λ_{j+1} − Γ_{+})(w^{+})^{2} + (λ_{j+1} − Γ_{−})(w^{−})^{2}] > 0 for all w ∈ N(L_{1} − λ_{j+1}I) \ {0} .

Suppose that for the ϵ > 0 and δ > 0 given in Lemma 2.2,

(G1) there is ρ > 0 such that for a.e. x ∈ Q, η(u) ∈ ℝ^{k}

γ_{+}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{+}(x) + ϵ if u > ρ
γ_{−}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{−}(x) + ϵ if u < −ρ

(G2) There are functions b(x) ∈ L_{∞}(Q) and k_{s}(x) ∈ L_{2}(Q) for each s > 0 such that

|g(x, u, η(u))| ≤ sb(x)( ∑_{|α|≤2m−1} |D^{α}u|^{2})^{1/2} + k_{s}(x), u ∈ V.

(F1) There are functions a(x) ∈ L_{∞}(Q) and d_{r}(x) ∈ L_{2}(Q) for each r > 0 such that

|f(x, ξ(u))| ≤ ra(x)( ∑_{|α|≤2m} |D^{α}u|^{2})^{1/2} + d_{r}(x), for all u ∈ V.

(F2) There is a constant k > 0 such that k ≤ c and

|f(x, η, ζ) − f(x, η, ζ′)| ≤ k ∑_{|α|=2m} |ζ_{α} − ζ′_{α}|

for a.e. x ∈ Q, all η ∈ ℝ^{k} and ζ, ζ′ ∈ R^{s′_{2m}} = R^{s_{2m}} − R^{s_{2m−1}}, where c is a constant in (3.1).

Then there is a u ∈ V that satisfies (3.2) for a.e. x ∈ Q. If k < c, then all other assertions of Theorem 3.1 also hold.

Proof. It is easy to see that (F) of Theorem 3.1 holds. Hence, it remains to verify (H) of that theorem, i.e. that H_{t} = B −tF is A-proper with respect to Γ_{B} for each t ∈ [0, 1) and H_{1} is pseudo A-proper. Since the embedding of V into H is compact, it suffices to show these facts for F_{t} = L − tF. Set B_{μ} = B − μI for some μ ≠ λ_{j} for each j. Then, for each t ∈ [0, 1], it follows from (F2), the Holder inequality, and an easy calculation that

(F_{t}u − F_{t}v, B_{μ}(u − v)) ≥ (1 − k/c)∥B_{μ}(u − v)∥^{2} + ϕ(u − v)  (3.22)

where the functional ϕ : V → R is given by

ϕ(u − v) = t(M(u, v) − M(v, v), B_{μ}(u − v)) + μ(u − v, B_{μ}(u − v)),

with M : V × V → H being the continuous form M(u, v) = f(x, η(u), ζ(v)). The functional ϕ is weakly continuous. Indeed, let u_{k} → u weakly in V. Then u_{k} → u in the W_{2}^{2m−1}-norm by the Sobolev imbedding theorem and by the results from [10], it is not hard to show that ϕ(u_{k} − u) → 0 as k → ∞. If k < c, then (F2) implies that F_{t} is A-proper with respect to Γ_{B} (see, e.g., in [16]-[18]). If k = c, then F_{t} is again A-proper for each t ∈ [0, 1) and it is easy to see that F_{1} is pseudo L_{μ}-monotone. Hence, F_{1} is pseudo A-proper with respect to Γ_{B} ([18]) and (H) of Theorem 3.1 holds.

Corollary 3.1 Let the conditions of Theorem 3.2 hold with (G1) replaced by

(G1’) γ_{±}(x) ≤ lim inf_{u→±∞} g(x, u, η(u)) ≤ lim sup_{u→±∞} g(x, u, η(u))

≤ Γ_{±}(t, x)

uniformly for a.e. (x, η(u)) ∈ Q × ℝ^{k}. Then there is a u ∈ V that satisfies (3.2) for a.e. x ∈ Q.

Proof. It is easy to see that (G1’) implies (G1).

4 Strong solvability of elliptic BVP’s

A. We shall apply the results of Section 3 to strong solvability of elliptic boundaryvalue problems in V of the form

∑_{|α|≤2m} A_{α}(x)D^{α}u(x) + g(x, u, Du, …, D^{2m−1}u)u + f(x, u, Du, …, D^{2m}) = h,

(4.1)

under non-uniform non-resonance conditions. Here Q ⊂ R^{n} is a bounded smooth domain, V is a closed subspace of W_{2}^{2m}(Q) containing the test functions, the linear part is elliptic and h ∈ L_{2}(Q). Assume the linear map L : V → L_{2}(Q), induced by the linear elliptic operator in (4.1), has closed range in H = L_{2}(Q) and satisfies conditions (L1), (B1) in Section 3 with B = −L + λ_{j}I. Here, L_{1} = L and A_{1} = 0.

Let λ ≠ λ_{j} for each j = 1, 2, … be fixed, Γ = {Y_{n}, Q_{n}} be a projectionally complete scheme for L_{2} and X_{n} = (B − λI)^{−1}(Y_{n}) ⊂ V for each n. Then Γ_{L} = {X_{n}, Y_{n}, Q_{n}} is an admissible or a projectionally complete scheme for (V, L_{2}). Since B − λI : V → L_{2} is linear, one-to-one and A-proper with respect to Γ_{L}, there is a constant c > 0 (depending) only on λ) such that

∥(B − λI)u∥ ≥ c∥u∥_{V} , u ∈ V.  (4.2)

Theorem 4.1 Let B = −L + λ_{j}I be a closed densely defined map in H such that R(B) = N(B)^{⊥}, (Bu, u) ≥ −r^{−1}∥Bu∥^{2} on V and if (Bu, u) = −r^{−1}∥Bu∥^{2} for some u ∈ V, then u ∈ N(−L + λ_{j}I) ⊕ N(−L + λ_{j+1}I). Suppose that there are functions γ_{±}, Γ_{±} ∈ L_{∞}(Q) such that for some j ∈ J ⊂ Z, one has

λ_{j} ≤ γ_{±}(x) ≤ Γ_{±}(x) ≤ λ_{j+1} for a.e. x ∈ Q

and

∫_{Q}[(γ_{+} − λ_{j})(v^{+})^{2} + (γ_{−} − λ_{j})(v^{−})^{2}] > 0 for all v ∈ N(L − λ_{j}I) \ {0}

and

∫_{Q}[(λ_{j+1} − Γ_{+})(w^{+})^{2} + (λ_{j+1} − Γ_{−})(w^{−})^{2}] > 0 for all w ∈ N(L − λ_{j+1}I) \ {0}.

Suppose that for ϵ > 0 and δ > 0 given in Lemma 3.2,

(G1) there is ρ > 0 such that for a.e. x ∈ Q

γ_{+}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{+}(x) + ϵ if u > ρ, η(u) ∈ ℝ^{k}
γ_{−}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{−}(x) + ϵ if u < −ρ, η(u) ∈ ℝ^{k}

(G2) There are functions b(x) ∈ L_{∞}(Q) and k_{s}(x) ∈ L_{2}(Q) for each s > 0 such that

|g(x, u, η(u))| ≤ sb(x)( ∑_{|α|≤2m−1} |D^{α}u|^{2})^{1/2} + k_{s}(x), u ∈ V.

(F) ∥F u∥ = ∥f(x, u, …, D^{2m}u)∥ ≤ β∥u∥_{V} + γ for some β ∈ (0, δ), γ > 0.

(H) H_{t} = L − tF : V → H is A-proper with respect to Γ_{L} for t ∈ [0, 1) and L − F is pseudo A-proper.

Then (4.1) has a solution u ∈ V for each h ∈ L_{2}. If L − F is A-proper, S(h) = (L − F)^{−1}({h}) is compact for each h ∈ L_{2} and card S(h) is constant, finite and positive on each connected component of the set L_{2} \ (L − F)(Σ).

Proof. It follows from Theorem 3.1 with L_{1} = L and A_{1} = 0. ♢ As before, we give now some concrete conditions on f, g so that (H) holds.

Theorem 4.2 Let B = −L + λ_{j}I be a closed densely defined map in H such that R(B) = N(B)^{⊥}, (Bu, u) ≥ −r^{−1}∥Bu∥^{2} on V and if (Bu, u) = −r^{−1}∥Bu∥^{2} for some u ∈ V, then u ∈ N(−L + λ_{j}I) ⊕ N(−L + λ_{j+1}I). Suppose that there are functions γ_{±}, Γ_{±} ∈ L_{∞}(Q) such that for some j ∈ J ⊂ Z, one has

λ_{j} ≤ γ_{±}(x) ≤ Γ_{±}(x) ≤ λ_{j+1} for a.e. x ∈ Q ,
∫_{Q}[(γ_{+} − λ_{j})(v^{+})^{2} + (γ_{−} − λ_{j})(v^{−})^{2}] > 0 for all v ∈ N(L − λ_{j}I) \ {0} ,

∫_{Q}[(λ_{j+1} − Γ_{+})(w^{+})^{2} + (λ_{j+1} − Γ_{−})(w^{−})^{2}] > 0 for all w ∈ N(L − λ_{j+1}I) \ {0}.

Furthermore, suppose that for the ϵ > 0 and δ > 0 given in Lemma 3.2, (G1) there is ρ > 0 such that for a.e. x ∈ Q

γ_{+}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{+}(x) + ϵ if u > ρ, η(u) ∈ ℝ^{k}
γ_{−}(x) − ϵ ≤ g(x, u, η(u)) ≤ Γ_{−}(x) + ϵ if u < −ρ, η(u) ∈ ℝ^{k}

(G2) There are functions b(x) ∈ L_{∞}(Q) and k_{s}(x) ∈ L_{2}(Q) for each s > 0 such that

|g(x, u, η(u))| ≤ sb(x)( ∑_{|α|≤2m−1} |D^{α}u|^{2})^{1/2} + k_{s}(x), u ∈ V.

(F1) There are functions a(x) ∈ L_{∞}(Q) and d_{r}(x) ∈ L_{2}(Q) for each r > 0 such that

|f(x, ξ(u))| ≤ ra(x)( ∑_{|α|≤2m} |D^{α}u|^{2})^{1/2} + d_{r}(x), for all u ∈ V.

(F2) There is a constant k > 0 such that k ≤ c and

|f(x, η, ζ) − f(x, η, ζ′)| ≤ k ∑_{|α|=2m} |ζ_{α} − ζ′_{α}|

for a.e. x ∈ Q, all η ∈ ℝ^{k} and ζ, ζ′ ∈ R^{s′_{2m}} = R^{s_{2m}} − R^{s_{2m−1}}, where c is a constant in (4.2).

Then there is a u ∈ V that satisfies (4.1) for a.e. x ∈ Q and all other assertions of Theorem 4.1 are valid if k < c.

Proof. It follows from Theorem 4.1 with L_{1} = L and A_{1} = 0. ♢ For our next result, we assume also

(L2) There is an integer j ≥ 1 such that λ_{j} < λ_{j+1} and Lw = λ_{k}w for k = j and k = j + 1, has the continuation property, that is if w(x) = 0 on a set of positive measure, then w(x) = 0 a.e. on Q.

Theorem 4.3 Let L satisfy (L1)-(L2) and (B1) with B = −L + λ_{j}I and let γ(x), Γ(x) ∈ L_{∞}(Q) be such that

(H1) λ_{j} ≤ γ(x) ≤ Γ(x) ≤ λ_{j+1} with meas{x ∈ Q|λ_{j} ≠ γ(x)} > 0 and meas{x ∈ Q|λ_{j+1} ≠ Γ(x)} > 0.

Suppose that (G1) of Theorem 3.4 holds and for ϵ > 0 and δ > 0 given by Lemma 3.2

(H2) γ(x) − ϵ ≤ g(x, ξ) ≤ Γ(x) + ϵ for all (x, ξ) ∈ Q × R^{s_{2m−1}}

(H3) ∥F u∥ = ∥f(x, u, …, D^{2m}u)∥ ≤ β∥u∥_{V} + γ for some β ∈ (0, δ), γ > 0.

(H4) H_{t} = L − tF is A-proper with respect to Γ_{L} for t ∈ [0, 1) and L − F is pseudo A-proper.

Then (4.1) has a solution u ∈ V and all other assertions of Theorem 4.1 are valid.

Proof. Clearly, (L2) and (H1) imply the integral inequalities in Theorem 4.2. Hence, the conclusion follows from this theorem.

Theorem 4.4 Let L and γ(x), Γ(x) be as in Theorem 4.3. Let f : Q × R^{s_{2m}} → R and g : Q × R^{s_{2m−1}} → R be Caratheodory functions such that

(F1) There are functions a(x) ∈ L_{∞}(Q) and d_{r}(x) ∈ L_{p}(Q) for each r > 0 such that

|f(x, ξ(u))| ≤ ra(x)( ∑_{|α|≤2m} |D^{α}u|^{2})^{1/2} + d_{r}(x), for all u ∈ V.

(F2) There is a constant k > 0 such that k ≤ c and

|f(x, η, ζ) − f(x, η, ζ′)| ≤ k ∑_{|α|=2m} |ζ_{α} − ζ′_{α}|

for a.e. x ∈ Q, all η ∈ ℝ^{k} and ζ, ζ′ ∈ R^{s′_{2m}} = R^{s_{2m}} − R^{s_{2m−1}}.

(G1) λ_{j} ≤ γ(x) ≤ liminf_{|u|→∞}g(x, u, η(u)) ≤ limsup_{|u|→∞}g(x, u, η(u)) ≤ Γ(x) ≤ λ_{j+1} uniformly for x ∈ Q and the non-u components η(u).

(G2) There are functions b(x) ∈ L_{∞}(Q) and k_{s}(x) ∈ L_{p}(Q) for each s > 0 such that

|g(x, u, η(u))| ≤ sb(x)( ∑_{|α|≤2m−1} |D^{α}u|^{2})^{1/2} + k_{s}(x), u ∈ V.

Then there is a u ∈ V that satisfies Eq. (4.1) for a.e. x ∈ Q and all other assertions of Theorem 4.1 are valid if k < c.

Proof. It follows from Theorem 4.2, as in the case of Corollary 3.1. ♢

Theorem 4.2 extends the existence result of Beresticki-de Figueiredo [3] who assumed f = 0 and g to depend only on u. A simplified proof of their results has been given by Mawhin [13]. If f does not depend on derivatives of order 2m, the existence part of Theorem 4.4 reduces to a result of Mawhin-Ward [14]. Their proofs are based on the Leray-Schauder and the coincidence degree theories respectively.

B. In this subsection we shall look at boundary value problems

Lu = λ_{1}u + g(x, u) = h, in Q, u|_{∂Q} = 0  (4.3)

where L is either selfadjoint or non-selfadjoint second order elliptic partial differential operator, and λ_{1} is the first (resp. principal) eigenvalue of the selfadjoint (resp. nonselfadjoint) operator −L, h ∈ L_{p}(Q) with p > n and g : Q×R → R is a Caratheodory function which grows at most linearly, i.e. there are a constant c_{1} > 0 and a function c_{2} ∈ L_{p}(Q), p > n, such that

|g(x, u)| ≤ c_{1}|u| + c_{2}(x)

for a.e. x ∈ Q and all u ∈ R. We assume that L is such that the Bony’s maximum principal (see eg. [4, 2]) and the abstract Krein-Rutman theorem [11] imply the existence of a real simple eigenvalue λ_{1} > 0 of

−Lu = λ_{1}u, u|_{∂Q} = 0

of minimal modulus such that there is a corresponding smooth eigenfunction w_{1} > 0 in Q and ∂w_{1}/∂η < 0 on ∂Q, where ∂/∂η stands for the outward normal derivative. Moreover, if L is nonseladjoint then λ_{1} is also an eigenvalue for the adjoint problem

−L^{∗}u = λ_{1}u, u|_{∂Q} = 0,

such that there is a corresponding smooth eigenfunction w_{1}^{∗} > 0 in Q and ∂w_{1}^{∗}/∂η < 0 on ∂Q.

Now, using Lemma 3.3, we shall prove the following existence result for (4.3) when the nonlinearity f(x, u) = λ_{1}u + g(x, u) ”lies” between the first two eigenvalues λ_{1} and λ_{2}. We assume, without loss of generality, that the following upper bounds are nonnegative

g_{+}(x) = lim sup_{u→∞} g(x, u)/u ≤ Γ_{+}(x), a.e. on Q (4.4)

g_{−}(x) = lim sup_{u→−∞} g(x, u)/u ≤ Γ_{−}(x), a.e. on Q. (4.5)

Since g grows linearly, we can suppose, without loss of generality, that Γ_{±} ∈ L_{p}(Q), p > n.

Theorem 4.5 Let g : Q × R → R be a Caratheodory function that grows linearly, g_{+}(x) and g_{−}(x) are different from zero on a set of nonzero measure, and

g(x, u)u ≥ 0  (4.6)

for a.e. x ∈ Q and all u ∈ R. Suppose that (4.4)-(4.5) hold and

0 ≤ Γ_{±}(x) ≤ r(= λ_{2} − λ_{1}), for a.e.x ∈ Q,  (4.7)

∫_{w>0}[r − Γ_{+}]w^{2}dx + ∫_{w<0}[r − Γ_{−}]w^{2}dx > 0, for all, w ∈ N(L + λ_{2}I) \ {0}.

(4.8)

Then Eq. (4.3) has at least one solution u ∈ W_{p}^{2}(Q) ∩ H_{0}^{1}(Q), p > n, for each h ∈ L_{p}(Q). Moreover, u ∈ C^{1,μ}(Q̄).

Proof. Let γ be a fixed constant with 0 < γ < r and define the operator E : W_{p}^{2}(Q) ∩ H_{0}^{1}(Q) ⊂ C^{1}(Q̄) → L_{p}(Q) by

Eu = Lu + λ_{1}u + ru .

We shall show that there exists a constant C > 0 independent of t such that ∥u∥_{C^{1}} ≤ C for all possible solutions u ∈ W_{p}^{2}(Q) ∩ H_{0}^{1}(Q) of the homotopy

H(t, u) = Lu + λ_{1}u + (1 − t)γu + tg(x, u) = th, t ∈ [0, 1).  (4.9)

Clearly, (4.9) has only the trivial solution for t = 0. If such a C does not exist, then there exist t_{k} ∈ (0, 1) and u_{k} ∈ W_{p}^{2}(Q) such ∥u_{k}∥ → ∞ and

Eu_{k} = t_{k}[γu_{k} − g(t_{k}, u_{k}) + h(x)], u|_{∂Q} = 0.  (4.10)

Set v_{k} = u_{k}/∥u_{k}∥_{C^{1}}. Then, (4.10) becomes

Ev_{k} = t_{k}[γv_{k} − g(x, u_{k})/∥u_{k}∥_{C^{1}} + h/∥u_{k}∥_{C^{1}}], v_{k}|_{∂Q} = 0.  (4.11)

We may assume that t_{k} → t and g(x, u_{k})/∥u_{k}∥_{C^{1}} ⇀ K(x) in L_{p}(Q) since g has a linear growth. Since g(x, u_{k})/∥u_{k}∥_{C^{1}} = g(x, u_{k})/u_{k}(x).v_{k}(x) ⇀ G(x)v(x) with G(x) ≠ 0 on a set of positive measure, we get that K(x) = G(x)v(x) ≠ 0 on a set of positive measure. Using L_{p}-estimate and the compact embedding of W_{p}^{2}(Q) into C^{1}(Q̄), we can deduce from (4.11) that v_{k} → v in C^{1}(Q̄), ∥v∥_{C^{1}} = 1 and v|_{∂Q} = 0. Moreover, {Lv_{k}} is also bounded in L_{p}(Q) by (4.11). Hence, by the reflexivity of L_{p} and the weak closedness of L, we may assume that Lv_{k} ⇀ Lv in L_{p} with v ∈ W_{p}^{2}(Q) ∩ H_{0}^{1}(Q) and v solves the equation

Ev = t[γv − K(x)], v|_{∂Q} = 0.  (4.12)

As in [9], Eq. (4.12) is equivalent to

Lv + λ_{1}v + p_{+}(x)v^{+} − p_{−}(x)v^{−} = 0, v|_{∂Q} = 0  (4.13)

where p_{+}(x) = (1 − t)γ + tk_{v}^{+}(x) and p_{−}(x) = (1 − t)γ + tk_{v}^{−}(x) and k_{v}(x) = K(x)/v(x) if v(x) ≠ 0 and k_{v} = 0 if v(x) = 0 since 0 ≤ k_{v}(x) ≤ Γ_{+}(x) if v(x) > 0 and 0 ≤ k_{v}(x) ≤ Γ_{−}(x) if v(x) < 0. Hence, by Lemma 3.3 (or Lemma 1 in [9]), we get that v ∈ N(L + λ_{1}I) \ {0}.

Next, passing to the limit in

(Lv_{k} + λ_{1}v_{k} + (1 − t)γv_{k} + t_{k}g(x, u_{k})/∥u_{k}∥_{C^{1}}, v_{k}) = (t_{k}h/∥u_{k}∥_{C^{1}}, v_{k})

we get

((1 − t)γv + tK, v) = 0.

Note that t ≠ 1, for otherwise (4.12)-(4.13) imply that p_{+}(x)v^{+} − p_{−}(x)v^{−} = 0 which leads to K(x) = 0 a.e. on Q, a contradiction. Hence, (K, v) < 0 since (1− t)γ∥v∥^{2} = −t(K, v). This contradicts the fact that 0 ≤ t_{k}(g(x, u_{k})/∥u_{k}∥_{C^{1}}, v_{k}) → t(K, v). Hence, we have shown that all solutions of (4.9) are bounded and, by the Leray-Schauder homotopy theorem, Eq (4.3) has a solution in W_{p}^{2}(Q) ∩ H_{0}^{1}(Q) for each p ∈ L_{p}(Q). ♢

Theorem 4.5 extends Theorem 1 in Iannacci-Nkashama-Ward [9] who showed the solvability of Eq (4.3) only for h ∈ L_{p}(Q) that are orthogonal to w_{1} but without assuming that g_{+}(x) and g_{−}(x) are not zero on a set of positive measure. On the other hand, their result extends some earlier ones of de Figueiredo and Ni [6], Gupta [7] and others. As in Theorems 4.4, it will be shown elsewhere that Theorem 4.5 can be extended to include nonlinearities depending on derivatives up to the second order.

5 Time periodic solutions of BVP’s for nonlinear parabolic and hyperbolic equations

The semi-abstract results in Section 3 have been used in [21] to prove the existence and the number of solutions of generalized periodic solutions (GPS), under nonuniform nonresonance conditions, for the nonlinear parabolic equation

u_{t} + A_{0}u + g(t, x, u, u_{t}, D_{x}u, …, D_{x}^{2m−1}u)u + f(t, x, u, u_{t}, D_{x}u, …, D_{x}^{2m}u) = h

in H = L_{2}(Ω), where Ω = [0, 2π] × Q with Q ⊂ R^{n}, A_{0} is a uniformly strongly elliptic operator of order 2m in x ∈ Q for each t ∈ [0, 2π], and the nonlinear hyperbolic equations with damping

σu_{t} + u_{tt} + A_{0}u = g(t, x, u, u_{t}, u_{tt}, D_{x}u, …, D_{x}^{2m−1}u)u
+f(t, x, u, u_{t}, u_{tt}, D_{x}u, …, D_{x}^{2m}u) + h

with h in H, σ ≠ 0, boundary conditions

u(t, .) ∈ H_{0}^{m}(Q) for all t ∈ (0, 2π) ,

and periodicity conditions

u(0, x) = u(2π, x) for all x ∈ Q .

These results extend the corresponding existence results in Nkashama-Willem [22], who assumed only the u dependence in g and f = 0 and used the coincidence degree theory.

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Petronije S. Milojević Department of Mathematical Sciences and CAMS New Jersey Institute of Technology Newark, NJ 07102, USA e-mail: pemilo@m.njit.edu