where Ω ⊂ ℝⁿ is open and bounded, h ∈ L²(Ω), λ_k is a simple eigenvalue of −Δ corresponding to the eigenvector ϕ_k, and the nonlinearity g : ℝ → ℝ satisfies the following conditions:

()

Landesman and Lazer [1] considered the problem (1.1)-(1.2) with continuous function g satisfying g(−∞) < g(ξ) < g(∞), where g(±∞) = lim _s→±∞g(s) exist and are finite. The authors showed that if ϕ_k is an eigenfunction corresponding to λ_k, Ω⁺ = {x ∈ Ω : ϕ_k > 0} and Ω⁻ = {x ∈ Ω : ϕ_k < 0}, then the necessary and sufficient condition for the existence of weak solution of (1.1)-(1.2) is that

()

The condition (1.3) is the well-known Landesman-Lazer condition, named after the authors. The result of the paper [1] has since been generalized by a number of authors which include [2–9], to mention a few.

We mention, briefly, few works without the assumption of the Landesman-Lazer condition. The perturbation of a second order linear elliptic problems by nonlinearity without Landesman-Lazer condition was investigated in [10]. The function g(u) was assumed to be a bounded continuous function satisfying

()

The nonhomogeneous term h was assumed to be an L^∞-function orthogonal to an eigenfunction ϕ in L², which corresponds to a simple eigenvalue λ₁. Ha [11] considered the solvability of an operator equation without the Landesman-Lazer condition. The author used a nonlinear Carathéodory function g(x, u) which satisfies the conditions

()

for almost all x ∈ Ω and all u ∈ ℝ, where b ∈ L²(Ω). The solvability of the operator equation is proved under some hypotheses on g(x, u). The nonhomogeneous term h was assumed to be an L²-function. Iannacci and Nkashama proved existence of solutions to a class of semilinear two-point eigenvalue boundary value problems at resonance without the Landesman-Lazer condition, by imposing the same conditions as in [11] in conjunction with some other hypotheses on g and h. Furthermore, the existence of solution was proved only for the eigenvalue λ = 1. Assuming a Carathéodory function f(x, u) with some growth restriction and assuming an L²-function h, Santanilla [12] proved existence of solution to a nonlinear eigenvalue boundary value problem (for eigenvalue λ = 1) without Landesman-Lazer condition. Du [13] proved the existence of solution for nonlinear second-order two-point boundary value problems, by allowing the eigenvalue λ of the problem to change near the eigenvalues of m²π² of the problem y^′′ + m²π²y = 0, y(0) = y(1) = 0. The author did not use the Landesman-Lazer condition and imposed weaker conditions on g(u) than in [12]. Recently, Sanni [14] proved the existence of solution to the same problem considered by Du [13] with λ = m²π² exactly, without assuming the Landesman-Lazer condition. The author assumed that |g^′(u)| ≤ C = constant and h ∈ L²(0,1). Other works without the assumption of Landesman-Lazer condition include [15–21]. We mention that most of the papers on this topic use the methods in [22] and [12]. The method of upper and lower solutions is used in [14]. For several other related resonance problems, we refer the reader to the book of Rădulescu [23].

The current work constitutes further deductions on the problem considered by Landesman and Lazer [1] and is motivated by previous works and by asking if it is possible to obtain a weak solution of (1.1)-(1.2) by setting u : = ϕ_kv(x). The answer is in the affirmative. The substitution gives rise to a degenerate semilinear elliptic equation. Consequently, we prove the existence of weak solution to the degenerate semilinear elliptic equation in a -weight Sobolev’s space, by using the Schaefer’s fixed point theorem. For information on weighted Sobolev’s spaces, the reader is referred to [24, 25]. The current work is significant in that the condition H enables a relaxation of the Landesman-Lazer condition (1.3), and the solution u to (1.1)-(1.2) is constructed using the eigenfunctions ϕ_k. Furthermore, the current analysis takes care of the situation where g(∞) = g(−∞) = 0.

The remaining part of this paper is organized as follows: the weighted Sobolev’s spaces used are defined in Section 2. In addition, we use the substitution u = ϕ_kv to get the degenerate semilinear elliptic equation in v, from which we give a definition of a weak solution. Furthermore, we state two theorems used in the proof of the existence result. In Section 3, we prove the existence and uniqueness of solution to an auxiliary linear problem. In Section 4, we prove a necessary condition for the existence of solution to (1.1)-(1.2) before proving the existence of solution to (1.1)-(1.2). At the end of Section 4, we prove that u : = ϕ_kv is in , provided that v ∈ X. Finally, we give an illustrative example in Section 5 for which our result applies.

2. Preliminaries

We define the following weighted Sobolev’s spaces used in this paper:

()

where

()

where

For brevity, we set .

Set u : = ϕ_kv(x) in (1.1) to deduce

()

Note that the first term on the left of (2.3) vanishes, multiply (2.3) by ϕ_k and use (1.2) to deduce

()

Thus, if we can prove the existence of solution to (2.4), then u : = ϕ_kv solves (1.1)-(1.2). Indeed, we will prove that the solution u belongs to the Sobolev space

Definition 2.1. We say that v ∈ X is a weak solution of the problem (2.4) provided

()

for each ζ ∈ X.

Definition 2.2. Let X be a Banach space and A : X → X a nonlinear mapping. A is called compact provided for each bounded sequence the sequence is precompact; that is, there exists a subsequence such that converges in X (see [26]).

The following theorems are applied in this paper.

Theorem 2.3 (Bolzano-Weierstrass). Every bounded sequence of real numbers has a convergent subsequence (see [27]).

Theorem 2.4 (Schaefer’s Fixed Point Theorem). Let X be a Banach space and

()

a continuous and compact mapping. Suppose further that the set

()

is bounded. Then A has a fixed point (see [26]).

3. Auxiliary Linear Problem

Consider the linear boundary value problem:

()

where μ is a strictly positive constant;

, g(ϕ_ks), and h are functions of x only.

Theorem 3.1 (a priori estimates). Let v be a solution of (3.1)-(3.2). Then v ∈ X and we have the estimate

()

for some appropriate constant C > 0.

Proof. Multiply (3.1) by v, integrate by parts and apply (3.2) to get

()

Using H, the second term in the bracket on the right side of (3.4) may be estimated as

()

Simplifying (3.5), we deduce

()

(see [26]) for some constant C = C(L, |g(0)|). Notice that (3.6) implies that

()

so that g(ϕ_ks) ∈ L²(Ω).

Using (3.7) and choosing ϵ > 0 sufficiently small in (3.4) and simplifying, we deduce (3.3).

Definition 3.2. (i) The bilinear form B[·, ·] associated with the elliptic operator L defined by (3.1) is

()

for v, ζ ∈ X,

(ii) v ∈ X is called a weak solution of the boundary value problem (3.1)-(3.2) provided

()

for all ζ ∈ X, where (·, ·) denotes the inner product in L²(Ω).

Theorem 3.3. B[u, v] satisfies the hypotheses of the Lax-Milgram theorem precisely. In other words, there exists constants α, β such that

(i)
|B[v, ζ]| ≤ α∥v∥_X∥ζ∥_X,
(ii)
,

for all v, ζ ∈ X.

Proof. We have

()

for appropriate constant α > 0. This proves (i).

We now proof (ii). We readily check that

()

for some constant β > 0. We can for example take β = min {1, μ}.

Theorem 3.4. There exists unique weak solution to the degenerate linear boundary value problem (3.1)-(3.2).

Proof. The hypothesis on h and (3.7) imply that g(ϕ_ks) − h ∈ L²(Ω). For fixed g(ϕ_ks) − h, set for all ζ ∈ X (where 〈, ·, 〉 denotes the pairing of X with its dual). This is a bounded linear functional on L²(Ω) and thus on X. Lax-Milgram theorem (see, e.g., [26]) can be applied to find a unique function v ∈ X satisfying

()

for all ζ ∈ X. Consequently, v is the unique weak solution of the problem (3.1)-(3.2).

4. Main Results

Theorem 4.1. The necessary condition that be a weak solution to (1.1)-(1.2) is that

()

Proof. Suppose is a weak solution of (1.1)-(1.2). For a test function ϕ_k, using integration by parts, we have:

()

from which (4.1) follows, since Δϕ_k + λ_kϕ_k = 0.

Theorem 4.2. Let the condition (4.1) of Theorem 4.1 holds. Then there exists a weak solution to the problem (2.4).

Proof. The proof is split in seven steps.

Step 1. A fixed point argument to (2.4) is

()

Define a mapping

()

by setting A[v] = w whenever w is derived from v via (4.3). We claim that A is a continuous and compact mapping. Our claim is proved in the next two steps.

Step 2. Choose , and define . For two solutions of (4.3), we have

()

Using (4.5), we obtain an analogous estimate to (3.4), namely:

()

Now

()

using the condition (H). We may now use (4.7) in (4.6) and simplify to deduce

()

for some constant C > 0. Thus, the mapping A is Lipschitz continuous, and hence continuous.

Step 3. Let be a bounded sequence in X. By Bolzano-Weierstrass theorem, it has a convergent subsequence, say . Define

()

Using (4.8)-(4.9), we deduce

()

Thus,

in X. Therefore, A is compact.

Step 4. Define a set K : = {p ∈ X : p = τA[p] for some 0 ≤ τ ≤ 1}. We will show that K is a bounded set. Let v ∈ K. Then v = τA[v] for some τ ∈ [0,1]. Thus, we have v/τ = A[v]. By the definition of the mapping A, w = v/τ is the solution of the problem

()

Now, (4.11) are equivalent to

()

Using (4.12) we have an analogous estimate to (3.3) of Theorem 3.1, namely:

()

Choosing τ ∈ [0,1] sufficiently small in (4.13) and simplifying, we conclude that

()

for some constant C > 0. Equation (4.14) implies that the set K is bounded, since v was arbitrarily chosen.

Since the mapping A is continuous and compact and the set K is bounded, by Schaefer’s fixed point theorem (see, e.g., [26]), the mapping A has a fixed point in X.

Step 5. Write . For m = 0,1, 2, …, inductively define v_m+1 ∈ X to be the unique weak solution of the linear boundary value problem

()

Clearly, our definition of v_m+1 ∈ X as the unique weak solution of (4.15)-(4.16) is justified by Theorem 3.4. Hence, by the definition of the mapping A, we have for m = 0,1, 2, …:

()

Since A has a fixed point in X, there exists v ∈ X such that

()

Step 6. Using (4.15)-(4.16), we obtain an analogous estimate to (3.3), namely:

()

for some appropriate constant C > 0. Using (4.18), we take the limit on the right side of (4.19) to deduce that

()

Equation (4.20) implies the existence of a subsequence

converging weakly in X to v ∈ X.

Furthermore, using (3.7), we deduce

()

Again, we use (4.18) to obtain the limit on the right side of (4.21) to deduce that

()

Equation (4.22) implies the existence of a subsequence

converging weakly in L²(Ω) to g(ϕ_kv) in L²(Ω).

Step 7. Finally, we verify that v is a weak solution of (2.4). For brevity, we take the subsequences of the last step as and . Fix ζ ∈ X. Multiply (4.15) by ζ, integrate by parts and apply (4.16) to get

()

Using the deductions of the last step, we let m → ∞ in (4.23) to obtain

()

from which canceling the terms in μ, we obtain (2.5) as desired.

Theorem 4.3. Let v ∈ X be the solution of (3.1)-(3.2). Then, the solution u : = ϕ_kv of (1.1)-(1.2) belongs to , and we have the estimate

()

for some constant C > 0.

Proof. We split the proof in two steps.

Step 1. Recall that ϕ_k satisfies the equations:

()

Multiplying (4.26) by v²ϕ_k, integrating by parts and applying (4.27) we compute

()

by Cauchy’s inequality with ϵ. Choosing ϵ > 0 sufficiently small in (4.29) and simplifying, we deduce

()

for some constant C > 0.

Step 2. We have

()

for some constant C > 0. Thus, u ∈ H¹(Ω). Hence, by a Sobolev’s embedding theorem (see [26, page 269]), we have that

, since u|_∂Ω = 0.

5. Illustrative Example

Consider the following special case for n = 1:

()

In this case, the eigenfunction ϕ_k = sin x, g(u) = −2u, and h = 1. Clearly g(u) is Lipschitz continuous and h ∈ L²(Ω). Provided the necessary condition

()

is satisfied; Theorems 4.2 and 4.3 ensure the existence of a solution

of the problem (5.1). Now, the problem (5.1) admits the solution

()

Using (5.3) in (5.2), it is not difficult to verify that the necessary condition

()

is satisfied.

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On the Weak Solution of a Semilinear Boundary Value Problem without the Landesman-Lazer Condition

Abstract

1. Introduction

2. Preliminaries

3. Auxiliary Linear Problem

4. Main Results

5. Illustrative Example

References

References

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On the Weak Solution of a Semilinear Boundary Value Problem without the Landesman-Lazer Condition

Abstract

1. Introduction

2. Preliminaries

3. Auxiliary Linear Problem

4. Main Results

5. Illustrative Example

References

References

Related

Information