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

Authors: Sikiru Adigun Sanni
Journal: International Journal of Differential Equations
 (2011)

---

## Abstract

We prove the existence of weak solution to a semilinear boundary value problem without the Landesman-Lazer condition.


---

## Body

## 1. Introduction

We consider the nonlinear boundary value problem(1.1)Δu+λku+g(u)=h(x)inΩ,(1.2)u=0on⁡∂Ω, where Ω⊂ℝn is open and bounded, h∈L2(Ω), λk is a simple eigenvalue of -Δ corresponding to the eigenvector ϕk, and the nonlinearity g:ℝ→ℝ satisfies the following conditions: (H)|g(u)-g(v)|≤L|u-v|(Lipschitzcontinuity)forsomeconstantL>0.Landesman and Lazer [1] considered the problem (1.1)-(1.2) with continuous function g satisfying g(-∞)<g(ξ)<g(∞), where g(±∞)=lims→±∞g(s) exist and are finite. The authors showed that if ϕk is an eigenfunction corresponding to λk, Ω+={x∈Ω:ϕk>0} and Ω-={x∈Ω:ϕk<0}, then the necessary and sufficient condition for the existence of weak solution of (1.1)-(1.2) is that(1.3)g(-∞)∫Ω+ϕkdx+g(∞)∫Ω-ϕkdx<∫Ωhϕkdx<g(∞)∫Ω+ϕkdx+g(-∞)∫Ω-ϕkdx.
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 (1.4)g(t)t≤0,t∈R.
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 (1.5)|g(x,u)|≤b(x),ug(x,u)≥0,
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:=ϕkv(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 ϕk2-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 ϕk. 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 Section2. In addition, we use the substitution u=ϕkv 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:=ϕkv is in H01(Ω), 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:(2.1)L2(Ω,ϕk2):={w:Ω⟶Rsuchthat‖w‖L2(Ω,ϕk2)<∞},
where ∥w∥L2(Ω,ϕk2)=∫Ωϕk2w2dx.(2.2)H1(Ω,ϕk2):={w:Ω⟶Rsuchthat‖w‖H1(Ω,ϕk2)<∞},
where ∥w∥H1(Ω,ϕk2)=∫Ωϕk2w2dx+∫Ωϕk2|∇w|2dx.For brevity, we setX=H1(Ω,ϕk2).Setu:=ϕkv(x) in (1.1) to deduce (2.3)-(Δϕk+λkϕk)v-ϕkΔv-2∇ϕk⋅∇v=g(ϕkv)-h(x)inΩ.
Note that the first term on the left of (2.3) vanishes, multiply (2.3) by ϕk and use (1.2) to deduce (2.4)-∇⋅(ϕk2∇v)=ϕkg(ϕkv)-ϕkh(x)inΩ,ϕkv=0on⁡∂Ω.
Thus, if we can prove the existence of solution to (2.4), then u:=ϕkv solves (1.1)-(1.2). Indeed, we will prove that the solution u belongs to the Sobolev space H01(Ω).Definition 2.1.
We say thatv∈X is a weak solution of the problem (2.4) provided
(2.5)∫Ωϕk2∇v⋅∇ζdx=∫Ωϕkζg(ϕkv)dx-∫Ωϕkζhdx,
for each ζ∈X.Definition 2.2.
LetX be a Banach space and A:X→X a nonlinear mapping. A is called compact provided for each bounded sequence {uk}k=1∞ the sequence {A[uk]}k=1∞ is precompact; that is, there exists a subsequence {ukj}j=1∞ such that {A[ukj]}j=1∞ 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).
LetX be a Banach space and
(2.6)A:X⟶X
a continuous and compact mapping. Suppose further that the set
(2.7){u∈X∣u=τA[u]forsome0≤τ≤1}
is bounded. Then A has a fixed point (see [26]).


## 3. Auxiliary Linear Problem

Consider the linear boundary value problem:(3.1)Lv:=-∇⋅(ϕk2∇v)+μϕk2v=μϕk2s+ϕkg(ϕks)-ϕkhinΩ,(3.2)ϕkv=0on⁡∂Ω,
where μ is a strictly positive constant; s∈L2(Ω,ϕk2), g(ϕks), and h are functions of x only.Theorem 3.1 (a priori estimates).
Letv be a solution of (3.1)-(3.2). Then v∈X and we have the estimate
(3.3)‖v‖X2≤C(‖s‖L2(Ω,ϕk2)2+‖h‖L2(Ω)2+1)<∞,
for some appropriate constant C>0.Proof.
Multiply (3.1) by v, integrate by parts and apply (3.2) to get
(3.4)∫Ωϕk2|∇v|2dx+μ∫Ωϕk2v2dx=μ∫Ωϕk2vsdx+∫Ωvϕkg(ϕks)dx-∫Ωvϕkhdx≤μ(∫Ωϕk2v2dx)1/2(∫Ωϕk2s2dx)1/2+(∫Ωϕk2v2dx)1/2(∫Ω|g(ϕks)|2dx)1/2+(∫Ωϕk2v2dx)1/2(∫Ωh2dx)1/2(byHölder'sinequality)≤3ϵ∫Ωϕk2v2dx+14ϵ(μ2∫Ωϕk2s2dx+∫Ω|g(ϕks)|2dx+∫Ωh2dx)(byCauchy'sinequalitywithϵ).
Using H, the second term in the bracket on the right side of (3.4) may be estimated as
(3.5)|g(ϕks)-g(0)|2≤L2|ϕks|2or|g(ϕks)|2≤-|g(0)|2+2|g(ϕks)||g(0)|+L2|ϕks|2≤-|g(0)|2+12|g(ϕks)|2+2|g(0)|2+L2|ϕks|2(byYoung'sinequality).
Simplifying (3.5), we deduce
(3.6)|g(ϕks)|≤C(1+|ϕks|),
(see [26]) for some constant C=C(L,|g(0)|). Notice that (3.6) implies that
(3.7)∫Ω|g(ϕks)|2dx≤C(1+‖s‖L2(Ω,ϕk2))2<∞,
so that g(ϕks)∈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 formB[·,·] associated with the elliptic operator L defined by (3.1) is
(3.8)B[v,ζ]:=∫Ωϕk2∇v⋅∇ζdx+μ∫Ωϕk2vζdx,
for v,ζ∈X,
(ii)v∈X is called a weak solution of the boundary value problem (3.1)-(3.2) provided
(3.9)B[u,ζ]=(μϕk2s+ϕkg(ϕks)-ϕkh,ζ),
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)
β∥v∥X2≤B[v,v], for all v,ζ∈X.Proof.
We have(3.10)|B[v,ζ]|=|∫Ωϕk2∇v⋅∇ζdx+μ∫Ωϕk2vζdx|≤μ(∫Ωϕk2v2dx)1/2(∫Ωϕk2ζ2dx)1/2+(∫Ωϕk2|∇v|2dx)1/2(∫Ωϕk2|∇ζ|2dx)1/2(byHölder′sinequality)≤α‖v‖X‖ζ‖X,
for appropriate constant α>0. This proves (i).
We now proof (ii). We readily check that(3.11)β‖v‖X2≤∫Ωϕk2|∇v|2dx+μ∫Ωϕk2v2dx=B[v,v],
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 onh and (3.7) imply that g(ϕks)-h∈L2(Ω). For fixed g(ϕks)-h, set 〈μϕk2s+ϕkg(ϕks)-ϕkh,ζ〉:=(μϕk2s+ϕkg(ϕks)-ϕkh,ζ)L2(Ω) 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
(3.12)B[v,ζ]=〈μϕk2s+ϕkg(ϕks)-ϕkh,ζ〉,
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 thatu∈H01(Ω) be a weak solution to (1.1)-(1.2) is that
(4.1)∫Ωg(u)ϕkdx=∫Ωhϕkdx.Proof.
Supposeu∈H01(Ω) is a weak solution of (1.1)-(1.2). For a test function ϕk, using integration by parts, we have:
(4.2)∫ΩΔuϕkdx+λk∫Ωuϕkdx+∫Ωg(u)ϕkdx=-∫Ω∇u⋅∇ϕk+λk∫Ωuϕkdx+∫Ωg(u)ϕkdx=∫Ωu(Δϕk+λkϕk)dx+∫Ωg(u)ϕkdx=∫Ωhϕkdx,
from which (4.1) follows, since Δϕk+λkϕk=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
(4.3)-∇⋅(ϕk2∇w)+μϕk2w=μϕk2v+ϕkg(ϕkv)-ϕkh(x)inΩ,ϕkw=0on⁡∂Ω.
Define a mapping
(4.4)A:X⟶X
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.
Choosev,ṽ∈X, and define A[v]=w,A[ṽ]=w̃. For two solutions w,w̃∈X of (4.3), we have
(4.5)-∇⋅[ϕk2∇(w-w̃)]+μϕk2(w-w̃)=μϕk2(v-ṽ)+ϕkg(ϕkv)-ϕkg(ϕkṽ)inΩ,ϕk(w-w̃)=0on⁡∂Ω.
Using (4.5), we obtain an analogous estimate to (3.4), namely:
(4.6)∫Ωϕk|∇w-∇w̃|2dx+μ∫Ωϕk2|w-w̃|2≤3ϵ∫Ωϕk2|w-w̃|2dx+14ϵ(μ2∫Ωϕk|v-ṽ|L2(Ω,ϕk2)2+∫Ω|g(ϕkv)-g(ϕkṽ)|2dx).
Now
(4.7)∫Ω|g(ϕkv)-g(ϕkṽ)|2dx≤∫Ωϕk2L2|v-ṽ|2dx,
using the condition (H). We may now use (4.7) in (4.6) and simplify to deduce
(4.8)‖A[v]-A[ṽ]‖X=‖w-w̃‖X≤C‖v-ṽ‖L2(Ω,ϕk2)≤C‖v-ṽ‖X,
for some constant C>0. Thus, the mapping A is Lipschitz continuous, and hence continuous.Step 3.
Let{vk}k=1∞ be a bounded sequence in X. By Bolzano-Weierstrass theorem, it has a convergent subsequence, say {vkj}j=1∞. Define
(4.9)v:=lim⁡kj→∞vkj.
Using (4.8)-(4.9), we deduce
(4.10)lim⁡kj→‖A[vkj]-A[v]‖X≤lim⁡kj→∞C‖vkj-v‖X=0.
Thus, A[vkj]→A[v] in X. Therefore, A is compact.Step 4.
Define a setK:={p∈X:p=τA[p]forsome0≤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
(4.11)-∇⋅[ϕk2∇(vτ)]+μϕk2vτ=μϕk2v+ϕkg(ϕkv)-ϕkh(x)inΩ,ϕkvτ=0on⁡∂Ω.
Now, (4.11) are equivalent to
(4.12)-∇⋅(ϕk2∇v)+μϕk2v=μτϕk2v+τϕkg(ϕkv)-τϕkh(x)inΩ,ϕkv=0on⁡∂Ω.
Using (4.12) we have an analogous estimate to (3.3) of Theorem 3.1, namely:
(4.13)‖v‖X2≤τC(‖v‖L2(Ω,ϕk2)2+‖h‖L2(Ω)2+1).
Choosing τ∈[0,1] sufficiently small in (4.13) and simplifying, we conclude that
(4.14)‖v‖X≤C‖h‖L2(Ω)2+1<∞
for some constant C>0. Equation (4.14) implies that the set K is bounded, since v was arbitrarily chosen.
Since the mappingA 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.
Writeϕkv0=ϕkv|∂Ω=0. For m=0,1,2,…, inductively define vm+1∈X to be the unique weak solution of the linear boundary value problem
(4.15)-∇⋅(ϕk2∇vm+1)+μϕk2vm+1=μϕk2vm+ϕkg(ϕkvm)-ϕkh(x)inΩ,(4.16)ϕkvm+1=0on⁡∂Ω.
Clearly, our definition of vm+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,…:(4.17)vm+1=A[vm].
Since A has a fixed point in X, there exists v∈X such that
(4.18)lim⁡m→∞vm+1=lim⁡m→∞A[vm]=A[v]=v.Step 6.
Using (4.15)-(4.16), we obtain an analogous estimate to (3.3), namely:
(4.19)‖vm+1‖X2≤C(‖vm‖L2(Ω,ϕk2)2+‖h‖L2(Ω)2+1)≤C(‖vm‖X2+‖h‖L2(Ω)2+1)
for some appropriate constant C>0. Using (4.18), we take the limit on the right side of (4.19) to deduce that
(4.20)sup⁡m‖vm‖X<∞.
Equation (4.20) implies the existence of a subsequence {vmj}j=1∞ converging weakly in X to v∈X.
Furthermore, using (3.7), we deduce
(4.21)∫Ω|g(ϕkvm)|2dx≤C(1+‖vm‖L2(Ω,ϕk2)2)2.
Again, we use (4.18) to obtain the limit on the right side of (4.21) to deduce that
(4.22)sup⁡m‖g(ϕkvm)‖L2(Ω)<∞.
Equation (4.22) implies the existence of a subsequence {g(ϕkvmj)}j=1∞ converging weakly in L2(Ω) to g(ϕkv) in L2(Ω).Step 7.
Finally, we verify thatv is a weak solution of (2.4). For brevity, we take the subsequences of the last step as {vm}m=1∞ and {g(ϕkvm)}m=1∞. Fix ζ∈X. Multiply (4.15) by ζ, integrate by parts and apply (4.16) to get
(4.23)∫Ωϕk2∇vm+1⋅∇ζdx+μ∫Ωϕk2vm+1ζdx=μ∫Ωϕk2vmζdx+∫Ωζϕkg(ϕkvm)dx-∫Ωζϕkhdx.
Using the deductions of the last step, we let m→∞ in (4.23) to obtain
(4.24)∫Ωϕk∇v⋅∇ζdx+μ∫Ωϕk2vζdx=μ∫Ωϕk2vζdx+∫Ωζϕkg(ϕkv)dx-∫Ωζϕkhdx,
from which canceling the terms in μ, we obtain (2.5) as desired.Theorem 4.3.
Letv∈X be the solution of (3.1)-(3.2). Then, the solution u:=ϕkv of (1.1)-(1.2) belongs to H01(Ω), and we have the estimate
(4.25)‖u‖H01(Ω)≤C‖v‖X,
for some constant C>0.Proof.
We split the proof in two steps.
Step 1.
Recall thatϕk satisfies the equations:
(4.26)Δϕk+λkϕk=0inΩ∈Rn,(4.27)ϕk=0on⁡∂Ω.
Multiplying (4.26) by v2ϕk, integrating by parts and applying (4.27) we compute
(4.28)∫Ωv2ϕkΔϕkdx+λk∫Ωv2ϕk2dx=0or∫Ω∇(v2ϕk)⋅∇ϕkdx=λk∫Ωv2ϕk2dxor∫Ω|∇ϕk|2v2dx=λk∫Ωv2ϕk2dx-2∫Ωϕkv∇v⋅∇ϕkdx(4.29)≤λk∫Ωv2ϕk2dx+ϵ∫Ω|∇ϕk|2v2dx+1ϵ∫Ωϕk2|∇v|2dx,
by Cauchy’s inequality with ϵ. Choosing ϵ>0 sufficiently small in (4.29) and simplifying, we deduce
(4.30)∫Ω|∇ϕk|2v2dx≤C‖v‖X2,
for some constant C>0.Step 2.
We have(4.31)∫Ωu2dx=∫Ωϕk2v2dx,∫Ω|∇u|2dx=∫Ω|∇(ϕkv)|2dx=∫Ω|∇ϕkv+ϕk∇v|2dx≤2∫Ω|∇ϕk|2v2dx+2∫Ωϕk2|∇v|2dx≤C‖v‖X2,(using(4.30))
for some constant C>0. Thus, u∈H1(Ω). Hence, by a Sobolev’s embedding theorem (see [26, page 269]), we have that u∈H01(Ω), since u|∂Ω=0.


## 5. Illustrative Example

Consider the following special case forn=1: (5.1)u′′+u-2u=1in(0,π),u(0)=u(π)=0.
In this case, the eigenfunction ϕk=sinx, g(u)=-2u, and h=1. Clearly g(u) is Lipschitz continuous and h∈L2(Ω). Provided the necessary condition(5.2)-2∫0πusin⁡xdx=∫0πsin⁡xdx
is satisfied; Theorems 4.2 and 4.3 ensure the existence of a solution u:=ϕkv(x)∈H01(Ω) of the problem (5.1). Now, the problem (5.1) admits the solution(5.3)u=sinh⁡(π-x)+sinh⁡xsinh⁡π-1.
Using (5.3) in (5.2), it is not difficult to verify that the necessary condition (5.4)-2∫0π(sinh⁡(π-x)+sinh⁡xsinh⁡π-1)sin⁡xdx=∫0πsin⁡xdx=2
is satisfied.



---

*Source: 801706-2011-11-30.xml*