ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ASYMMETRIC SUPERLINEAR PROBLEMS UNDER STRONG RESONANCE CONDITIONS

LEANDRO RECOVA, ADOLFO RUMBOS

Abstract. We study the existence and multiplicity of solutions of the problem

−∆u=−λ1u^{−}+g(x, u), in Ω;

u= 0, on∂Ω, (1)

where Ω is a smooth bounded domain inR^{N}(N≥2),u^{−}denotes the negative
part ofu: Ω→R,λ1 is the first eigenvalue of theN-dimensional Laplacian
with Dirichlet boundary conditions in Ω, andg: Ω×R→Ris a continuous
function with g(x,0) = 0 for all x ∈ Ω. We assume that the nonlinearity
g(x, s) has a strong resonant behavior for large negative values of sand is
superlinear, but subcritical, for large positive values ofs. Because of the lack
of compactness in this kind of problem, we establish conditions under which
the associated energy functional satisfies the Palais-Smale condition. We prove
the existence of three nontrivial solutions of problem (1) as a consequence of
Ekeland’s Variational Principle and a variant of the mountain pass theorem
due to Pucci and Serrin [14].

1. Introduction

Let Ω denote a bounded, connected, open subset ofR^{N}, forN >2, with smooth
boundary ∂Ω. We are interested in the existence and multiplicity of solutions of
the semilinear elliptic boundary value problem (BVP):

−∆u=−λ1u^{−}+g(x, u), in Ω;

u= 0, on∂Ω, (1.1)

whereu^{−}denotes the negative part ofu: Ω→R,λ1is the first eigenvalue of theN-
dimensional Laplacian with Dirichlet boundary conditions in Ω, andg: Ω×R→R
and its primitive

G(x, s) = Z s

0

g(x, ξ)dξ, forx∈Ω ands∈R, (1.2) satisfy the following conditions:

(A1) g∈C(Ω×R,R) andg(x,0) = 0 for all x∈Ω.

(A2) lim_{s→−∞}g(x, s) = 0, uniformly for a.e. x∈Ω.

2010Mathematics Subject Classification. 35J20.

Key words and phrases. Strong resonance; Palais-Smale condition; Ekeland’s principle.

c

2017 Texas State University.

Submitted February 19, 2017. Published June 23, 2017.

1

(A3) There exists a constant σsuch that 1 6σ <(N+ 2)/(N−2) forN >3, or 16σ <∞forN = 2, and

s→+∞lim g(x, s)

s^{σ} = 0,
uniformly for a.e. x∈Ω.

(A4) There are constantsµ >max

2,_{N}^{2N σ}_{+2} andso>0 such that
0< µG(x, s)6sg(x, s), fors>so andx∈Ω.

(A5) lim_{s→−∞}G(x, s)≡G_{−∞}, uniformly inx, where G_{−∞}∈R.
Writing

q(x, s) =−λ1s^{−}+g(x, s), for (x, s)∈Ω×R, (1.3)
we assume further that

(A6) q∈C^{1}(Ω×R,R) andq(x,0) = 0; and
(A7) ^{∂q}_{∂s}(x,0) =a, for allx∈Ω, wherea > λ1.

We determine conditions under which the BVP in (1.1) has nontrivial solutions.

By a solution of (1.1) we mean a weak solution; i.e, a functionu∈H_{0}^{1}(Ω) satisfying
Z

Ω

∇u· ∇v dx+λ1

Z

Ω

u^{−}v dx−
Z

Ω

g(x, u)v dx= 0, for allv∈H_{0}^{1}(Ω), (1.4)
where H_{0}^{1}(Ω) is the Sobolev space obtained through completion of C_{c}^{∞}(Ω) with
respect to the metric induced by the norm

kuk=Z

Ω

|∇u|^{2}dx1/2

, for allu∈H_{0}^{1}(Ω).

The weak solutions of (1.1) are the critical points of the functionalJ:H_{0}^{1}(Ω)→
Rgiven by

J(u) = 1 2 Z

Ω

|∇u|^{2}dx−λ_{1}
2

Z

Ω

(u^{−})^{2}dx−
Z

Ω

G(x, u(x))dx, (1.5)
for u ∈ H_{0}^{1}(Ω). Indeed, the functionalJ given in (1.5) is in C^{1}(H_{0}^{1}(Ω),R) with
Fr´echet derivative at everyu∈H_{0}^{1}(Ω) given by

J^{0}(u)v=
Z

Ω

∇u· ∇v dx+λ1

Z

Ω

u^{−}v dx−

Z

Ω

g(x, u)v dx, for allv∈H_{0}^{1}(Ω). (1.6)
Thus, comparing (1.4) with (1.6), we see that critical points ofJ are weak solutions
of (1.1).

In many problems, the following condition, known as the Palais-Smale condition, is usually needed to prove the existence of critical points of a functional.

Definition 1.1(Palais-Smale Sequence). LetJ ∈C^{1}(X,R), whereX is a Banach
space with normk · k. A sequence (um) in X satisfying

J(um)→c and kJ^{0}(um)k →0 asm→ ∞,
is said to be a Palais-Smale sequence forJ at c.

If (u_{m}) is a sequence satisfying

(i) |J(um)|6M for allm= 1,2,3, . . .and someM >0;

(ii) kJ^{0}(um)k →0 asm→ ∞,

we say that (um) is a Palais-Smale sequence forJ.

Definition 1.2(Palais-Smale Condition). A functionalJ ∈C^{1}(X,R), whereX is
a Banach space with normk · k, is said to satisfy the the Palais-Smale condition
at c, denoted (PS)_{c}, if every Palais-Smale sequence for J at c has a convergent
subsequence. In particular, if J has a Palais-Smale sequence atc, and J satisfies
the (PS)_{c} condition, thencis a critical value ofJ.

We say that J satisfies the (PS) condition if every (PS) sequence for J has a convergent subsequence.

It follows from condition (A2) and (1.3) that

s→−∞lim q(x, s)

s =λ_{1}, for allx∈Ω. (1.7)

The condition in (1.7) makes the BVP in (1.1) into a problem at resonance. Exis- tence for problems at resonance is sometimes obtained by imposing a Landesman- Lazer type condition on the nonlinearity. The authors of this article obtained existence and multiplicity for the BVP (1.1) in [15] for the case in which

s→−∞lim g(x, s) =g_{−∞}(x)
exists for allx∈Ω, and

Z

Ω

g_{−∞}(x)ϕ_{1}(x)dx >0, (1.8)
whereϕ1is an eigenfunction of theN-dimensional Laplacian over Ω corresponding
to the eigenvalue λ1, with ϕ1(x) > 0 for all x ∈ Ω. In the case in which the
Landesman-Lazer condition (1.8) holds, the authors were able to prove that the
functionalJ defined in (1.5) satisfies the (PS) condition.

Note that the assumption in (A2) prevents condition (1.8) from holding true.

So that, a Landesman-Lazer type condition does not hold for the problem at hand.

As a consequence, we will not be able to prove that the functional J satisfies
the (PS) condition. We will, however, be able to show that J satisfies the (PS)_{c}
condition at values ofcthat are not in an exceptional set, Λ. In the case in which
conditions (A1)–(A5) hold true, we will prove in the next section that the functional
J ∈C^{1}(H_{0}^{1}(Ω),R) given in (1.5) satisfies the (PS)_{c} condition provided that

c6=−G_{−∞}|Ω|, (1.9)

where|Ω|denotes the Lebesgue measure of Ω; thus, the exceptional set in this case is

Λ ={−G_{−∞}|Ω|}. (1.10)

It is not hard to see that the functionalJ defined in (1.5) does not satisfy the (PS)_{c}
condition atc =c−∞ ≡ −G−∞|Ω|. Indeed, the sequence of functions (um) given
by

um=−mϕ1, form= 1,2,3, . . . ,
is a (PS)_{c}

−∞ sequence, as a consequence of assumptions (A2) and (A5). However,
ku_{m+1}−u_{m}k=kϕ_{1}k, for allm= 1,2,3, . . .;

so that (um) has no convergent subsequence.

This lack of compactness is typical of problems at strong resonance. The term strong resonance refers to the situation described by the assumptions in (A2) and (A5) and was introduced by Bartolo, Benci and Fortunato in [3]. In [3], the authors

consider problems similar to (1.1) in whichg is bounded, and in which the excep- tional set is a singleton as in (1.10); more precisely, the authors of [3] consider the class of BVPs of the form

−∆u=qk(u), in Ω;

u= 0, on∂Ω, (1.11)

where

q_{k}(s) =λ_{k}s−g(s),

with λk an eigenvalue of the Laplacian, and g:R → R a bounded, continuous function with

lim

|s|→∞sg(s) = 0.

Furthermore, the authors of [3] assume that the function G(s) =

Z s

−∞

g(ξ)dξ

is defined for alls∈R, and satisfiesG(s)>0 for alls∈R, and

s→∞lim G(s) = 0.

The authors of [3] proved existence of weak solutions of BVP (1.11) by introducing a compactness condition (Condition (C)) that replaces the (PS) condition, and using the new condition to prove a variant of the deformation lemma.

In [7] and [8], Costa and Silva are able to obtain some of the existence and
multiplicity results of Bartolo, Benci and Fortunato [3] by establishing that the
associated functionalJsatisfies the (PS)_{c}condition for values ofcthat are not in an
exceptional set. More recently, Hirano, Li and Wang [12] have used Morse Theory
to obtain multiplicity results for this type of problems with strong resonance. In
[12], the exceptional set, Λ, consists of a finite number of values. They are able
to compute critical groups around the values in Λ; that is, critical groups are
computed at values where the (PS) condition fails. These critical groups are then
incorporated into a new version of the Morse inequality, which allowed the authors
of [12] to obtain multiplicity results.

In all the articles cited so far, the nonlinearitygis assumed to be bounded. In the present work, we relax that assumption by allowing g(x, s) to grow superlinearly, but subcritically, ins, for positive values ofs(see (A3) and (A4)), whileg(x, s) is bounded for negative values ofs(see (A2)).

For additional information on problems at strong resonance in the context of critical point theory, the reader is referred to the works of Arcoya and Costa [2], Li [13], and Chang and Liu [6], and the bibliographies found in those papers.

After establishing that the functionalJ defined in (1.5) satisfies the (PS)_{c}condi-
tion forc6=−G_{−∞}|Ω|in Section 2, under assumptions (A1)–(A6), we then proceed
to show in Section 3 that J has a local minimizer distinct from 0, provided that
(A5) holds with G_{−∞} 6 0, and (A7) also holds. In subsequent sections, we in-
troduce an additional condition on the nonlinearity that will allow us to prove the
existence of more critical point ofJ. In particular, we will assume the following:

(A8) there existss_{1}>0 such thatg(x, s_{1}) = 0 for allx∈Ω.

In Section 4, we prove that if, in addition to (A1)–(A5), withG−∞ 60, (A6) and (A7), we also assume (A8), then J has a second local minimizer distinct from 0.

Finally, in Section 5, we prove the existence of a third nontrivial critical point ofJ

by means of a variant of the mountain-pass theorem proved by Pucci and Serrin in [14].

2. Proof of the Palais-Smale Condition

In this section we prove that the functional J defined in (1.5), where g and
its primitive G satisfy the conditions in (A1)–(A5), satisfies the (PS)_{c} condition
provided thatc6=−G_{−∞}|Ω|.

Proposition 2.1. Assume that g and G satisfy (A1)–(A5), and define J as in
(1.5). Then,J satisfies the(P S)_{c} forc6=−G_{−∞}|Ω|.

Proof. Assume thatc6=−G−∞|Ω|and let (um) be a sequence inH_{0}^{1}(Ω) satisfying
J(um)→c and kJ^{0}(um)k →0 asm→ ∞. (2.1)
Thus, according to (1.5) and (1.6),

1 2

Z

Ω

|∇um|^{2}dx−λ_{1}
2

Z

Ω

(u^{−}_{m})^{2}dx−
Z

Ω

G(x, u_{m}(x))dx→c, as m→ ∞, (2.2)
and

Z

Ω

∇um· ∇v dx+λ1

Z

Ω

u^{−}_{m}v dx−
Z

Ω

g(x, um)v dx

6εmkvk, (2.3)
for all m and all v ∈ H_{0}^{1}(Ω), where (εm) is a sequence of positive numbers that
tends to 0 asm→ ∞.

We will show that (um) has a subsequence that converges inH_{0}^{1}(Ω). It follows
from (A2) that there existss_{1}>0 such that

−16g(x, s)61, fors <−s1, and allx∈Ω. (2.4) Consequently,

− |s|6sg(x, s)6|s|, fors <−s1, and allx∈Ω, (2.5) and

−C1− |s|6G(x, s)6C1+|s|, fors <−s1, and allx∈Ω, (2.6)
for some positive constantC_{1}. Combining (2.5) and (2.6), and using the continuity
ofg, we can find a positive constantC_{2}such that

−C_{2}−3|s|6sg(x, s)−2G(x, s)6C_{2}+ 3|s|, fors60, and allx∈Ω. (2.7)
Similarly, we obtain from (A3) that there exists a positive constantC_{3}such that

|g(x, s)|6C3+|s|^{σ}, fors>0 andx∈Ω. (2.8)
Finally, we obtain from (A4) that there exist positive constantsC4andC5such
that

G(x, s)>C_{4}s^{µ}−C_{5}, fors>0 andx∈Ω. (2.9)
Now, it follows from (2.2) that there exists a positive constantC_{6}such that

Z

Ω

|∇um|^{2}dx−λ1

Z

Ω

(u^{−}_{m})^{2}dx−
Z

Ω

2G(x, um(x))dx

6C6, for allm. (2.10)
Takingv=u_{m}in (2.3), we obtain

Z

Ω

|∇um|^{2}dx−λ1

Z

Ω

(u^{−}_{m})^{2}dx−
Z

Ω

g(x, um(x))um(x)dx

6εmkumk, (2.11) for allm.

Combining (2.10) and (2.11) we then obtain that

Z

Ω

[g(x, um(x))um(x)−2G(x, um(x))]dx

6C6+εmkumk, for allm. (2.12) Next, define the sets

Ω^{−}_{m}={x∈Ω|um(x)<0}; Ω^{+}_{m}={x∈Ω|um(x)>0};

Ω^{o}_{m}={x∈Ω|06um(x)6so}; Ω^{s}_{m}^{o} ={x∈Ω|um(x)> so}.

Then, using the estimate in (2.7),

Z

Ω^{−}_{m}

[g(x, um(x))um(x)−2G(x, um(x))]dx

6C+ 3ku^{−}_{m}kL^{1}, for allm. (2.13)
Note: From this point on in this paper, the symbolCwill be used to represent any
positive constant. Thus,Cmight represent different constants in various estimates,
even within the same inequality.

It follows from (2.12) and (2.13) that

Z

Ω^{+}m

[g(x, um(x))um(x)−2G(x, um(x))]dx

6C+εmkumk+ 3ku^{−}_{m}k_{L}1, (2.14)
for allm.

Using the continuity of g andGwe deduce the existence of a positive constant C such that

Z

Ω^{o}_{m}

[g(x, u_{m}(x))u_{m}(x)−2G(x, u_{m}(x))]dx6C, for allm. (2.15)
On the other hand, using (A4) we obtain that

(µ−2) Z

Ω^{so}_{m}

G(x, u_{m}(x))dx6
Z

Ω^{so}_{m}

[g(x, u_{m}(x))u_{m}(x)−2G(x, u_{m}(x))]dx,
for allm; so that, using this estimate in conjunction with (2.15), (2.13), (2.12) and
the assumption thatµ >2, we obtain that

Z

Ω^{so}_{m}

G(x, u_{m}(x))dx6C+ 3ku^{−}_{m}k_{L}1+ε_{m}ku_{m}k, for allm. (2.16)
Noting that Ω^{+}_{m}= Ω^{o}_{m}∪Ω^{s}_{m}^{o}, we obtain from (2.16) that

Z

Ω^{+}m

G(x, um(x))dx

6C+ 3ku^{−}_{m}k_{L}1+εmkumk, for allm. (2.17)
Next, we takev=−u^{−}_{m}in (2.3) to obtain

Z

Ω

|∇u^{−}_{m}|^{2}dx−λ_{1}
Z

Ω

(u^{−}_{m})^{2}dx−
Z

Ω^{−}_{m}

g(x, u_{m})u_{m}dx

6ε_{m}ku^{−}_{m}k, (2.18)
We get from (2.5) and (2.6) that

Z

Ω^{−}_{m}

g(x, u_{m}(x))u_{m}(x)dx

6C+ku^{−}_{m}k_{L}1, for allm, (2.19)
and

Z

Ω^{−}_{m}

G(x, um(x))dx

6C+ku^{−}_{m}k_{L}1, for allm. (2.20)

Takingv=u^{+}_{m}in (2.3) we then get

Z

Ω

|∇u^{+}_{m}|^{2}dx−
Z

Ω^{+}_{m}

g(x, u_{m})u_{m}dx

6ε_{m}ku^{+}_{m}k, for allm. (2.21)
It follows from (2.21), (2.17) and (2.14) that

Z

Ω

|∇u^{+}_{m}|^{2}dx6C+ε_{m}ku^{+}_{m}k+ 2ε_{m}ku_{m}k+ 6ku^{−}_{m}k_{L}1, for allm,
which can be rewritten as

Z

Ω

|∇u^{+}_{m}|^{2}dx6C+ 3εmku^{+}_{m}k+ 2εmku^{−}_{m}k+ 6ku^{−}_{m}k_{L}1, for allm, (2.22)
by the triangle inequality.

We claim that, if (1.9) holds true, then (u^{−}_{m}) is bounded. We argue by contra-
diction. Suppose, passing to a subsequence if necessary, that

ku^{−}_{m}k → ∞, as m→ ∞. (2.23)
It follows from (2.22), the Cauchy-Schwarz inequality, and the Poincar´e inequality
that

ku^{+}_{m}k6C+C
q

1 +ku^{−}mk, for allm. (2.24)
Combining (2.24) and (2.23) we then deduce that

m→∞lim
ku^{+}_{m}k

ku^{−}mk = 0. (2.25)

Next, define

vm=− u^{−}_{m}

ku^{−}mk, for allm; (2.26)

so thatkv_{m}k= 1 for allm. We may therefore extract a subsequence (v_{m}_{k}) of (v_{m})
such that

vm_{k} * v (weakly) ask→ ∞, (2.27)

for some v ∈ H_{0}^{1}(Ω). We may also assume, passing to further subsequences if
necessary, that

vm_{k} →vinL^{2}(Ω) as k→ ∞, (2.28)
v_{m}_{k}(x)→v(x) for a.e. x∈Ω ask→ ∞. (2.29)
Now, it follows from (2.3) and the fact thatum_{k}=u^{+}_{m}_{k}−u^{−}_{m}_{k} that

−

Z

Ω

∇u^{−}_{m}

k· ∇v dx+λ1

Z

Ω

u^{−}_{m}

kv dx
6εm_{k}kvk+

Z

Ω

|∇u^{+}_{m}_{k}· ∇v|dx+
Z

Ω

|g(x, um_{k}(x))||v|dx,

(2.30)

for allk and allv ∈H_{0}^{1}(Ω). Using the Cauchy-Schwarz inequality, we can rewrite
the estimate in (2.30) as

Z

Ω

∇u^{−}_{m}

k· ∇v dx−λ1

Z

Ω

u^{−}_{m}

kv dx
6εm_{k}kvk+ku^{+}_{m}_{k}kkvk+

Z

Ω

|g(x, um_{k}(x))||v|dx,

(2.31)

for allkand allv∈H_{0}^{1}(Ω).

Next, we estimate the last integral on the right-hand side of (2.31) by first writing Z

Ω

|g(x, u_{m}_{k}(x))||v|dx

= Z

Ω^{−}_{mk}

|g(x, u_{m}_{k}(x))||v|dx+
Z

Ω^{+}_{mk}

|g(x, u_{m}_{k}(x))||v|dx,

(2.32)

for allkand allv∈H_{0}^{1}(Ω).

To estimate the first integral on the right-hand side of (2.32), we use (2.4), the Cauchy-Schwarz inequality, and the Poincar´e inequality to get that

Z

Ω^{−}_{mk}

|g(x, umk(x))||v|dx

6Ckvk, for allk and allv∈H_{0}^{1}(Ω). (2.33)
To estimate the second integral in the right-hand side of (2.32), apply H¨older’s
inequality with p= 2N/(N + 2) and q= 2N/(N−2) forN >3. If N = 2, take
16p6µ/σ, which can be done because (A4) implies thatpσ < µ. Then,

Z

Ω^{+}_{mk}

|g(x, umk)||v|dx 6Z

Ω^{+}_{mk}

|g(x, umk)|^{p}^{1/p}Z

Ω

|v|^{q}^{1/q}

; so that, in view of (2.8) and the Sobolev embedding theorem,

Z

Ω^{+}_{mk}

|g(x, um_{k})||v|dx

6CZ

Ω

(C+|u^{+}_{m}_{k}|^{σ})^{p}dx^{1/p}

kvk, (2.34)
for allkand allv∈H_{0}^{1}(Ω). We then obtain from (2.34) and Minkowski’s inequality
that

Z

Ω^{+}_{mk}

|g(x, um_{k})||v|dx

6C(1 +ku^{+}_{m}_{k}k^{σ}_{L}pσ)kvk, (2.35)
for allkand allv∈H_{0}^{1}(Ω).

Combining (2.32) with the estimates in (2.33) and (2.35), we then obtain that Z

Ω

|g(x, um_{k}(x))||v|dx6C(1 +ku^{+}_{m}_{k}k^{σ}_{L}pσ)kvk, (2.36)
for allkand allv∈H_{0}^{1}(Ω).

Finally, combining the estimates in (2.31) and (2.36),

Z

Ω

∇u^{−}_{m}_{k}· ∇v dx−λ1

Z

Ω

u^{−}_{m}_{k}v dx

6C(1 +ku^{+}_{m}_{k}k+ku^{+}_{m}_{k}k^{σ}_{L}pσ)kvk, (2.37)
for allkand allv∈H_{0}^{1}(Ω).

Next, divide on both sides of (2.37) byku^{−}_{m}_{k}kand use (2.26) to get

Z

Ω

∇vm_{k}· ∇v dx−λ1

Z

Ω

vm_{k}v dx
6C 1

ku^{−}mkk+ku^{+}_{m}_{k}k

ku^{−}mkk +ku^{+}_{m}_{k}k^{σ}_{L}pσ

ku^{−}mkk
kvk,

(2.38)

for allkand allv∈H_{0}^{1}(Ω).

We will show next that

lim

k→∞

ku^{+}_{m}_{k}k^{σ}_{L}pσ

ku^{−}mkk = 0. (2.39)

Using the estimates in (2.9) and (2.17) we obtain Z

Ω

(u^{+}_{m}_{k})^{µ}dx6C+Cku^{−}_{m}_{k}k+εm_{k}kum_{k}k, for allk, (2.40)
where we have also used the Cauchy-Schwarz and Poincar´e inequalities. It then
follows from (2.40) that

ku^{+}_{m}_{k}kL^{µ} 6C(1 +ku^{−}_{m}_{k}k^{1/µ}+kum_{k}k^{1/µ}), for allk. (2.41)
Next, dividing on both sides of (2.41) byku^{−}_{m}_{k}k^{1/σ} and using the fact that

kumkk6ku^{+}_{m}_{k}k+ku^{−}_{m}_{k}k
we obtain

ku^{+}_{m}_{k}kL^{µ}

ku^{−}mkk^{1/σ} 6C 1

ku^{−}mkk^{1/σ} +ku^{−}_{m}_{k}k^{1/µ}

ku^{−}mkk^{1/σ} +ku^{+}_{m}_{k}k^{1/µ}
ku^{−}mkk^{1/σ}

, for allk, which we can rewrite as

ku^{+}_{m}_{k}kL^{µ}

ku^{−}m_{k}k^{1/σ} 6C 1

ku^{−}m_{k}k^{1/σ}+ 1

ku^{−}m_{k}k^{1/σ−1/µ}+ku^{+}_{m}_{k}k
ku^{−}m_{k}k

1/µ 1
ku^{−}m_{k}k^{1/σ−1/µ}

, (2.42) for allk. Now, in view of (A4) we see thatµ > σ; we then obtain from (2.23) and (2.25), in conjunction with (2.42), that

k→∞lim
ku^{+}_{m}

kkL^{µ}

ku^{−}m_{k}k^{1/σ} = 0. (2.43)

Next, using the condition µ > pσ in (A4) to apply H¨older’s inequality with
p_{1}=µ/pσandp_{2}its conjugate exponent we obtain

ku^{+}_{m}_{k}k^{pσ}_{L}pσ=
Z

Ω

(u^{+}_{m}_{k})^{pσ}dx6
Z

Ω

(u^{+}_{m}_{k})^{µ}dxpσ/µ

|Ω|^{1/p}^{2};
so that,

ku^{+}_{m}_{k}k^{σ}_{L}pσ6Cku^{+}_{m}_{k}k^{σ}_{L}µ, for allk,
and, dividing on both sides byku^{−}_{m}

kk,
ku^{+}_{m}_{k}k^{σ}_{L}pσ

ku^{−}mkk 6Cku^{+}_{m}_{k}kL^{µ}

ku^{−}mkk^{1/σ}
^{σ}

, for allk. (2.44)

It then follows from (2.43) and (2.44) that

k→∞lim

ku^{+}_{m}_{k}k^{σ}_{L}pσ

ku^{−}mkk = 0,
which is (2.39).

Using (2.23), (2.25) and (2.39), we obtain from (2.38) that

k→∞lim Z

Ω

∇vm_{k}· ∇v dx−λ1

Z

Ω

vm_{k}v dx

= 0, for allv∈H_{0}^{1}(Ω). (2.45)
It then follows from (2.26), (2.27) and (2.45) that

Z

Ω

∇v· ∇v dx−λ_{1}
Z

Ω

vv dx= 0, for allv∈H_{0}^{1}(Ω);

so that,v is a weak solution of the BVP

−∆u=λ1u, in Ω;

u= 0, on∂Ω. (2.46)

Now, it follows from (2.18) that

Z

Ω

|∇u^{−}_{m}_{k}|^{2}dx−λ1

Z

Ω

(u^{−}_{m}_{k})^{2}dx

6εm_{k}ku^{−}_{m}_{k}k+
Z

Ω^{−}_{mk}

g(x, um_{k})um_{k}dx
, (2.47)
for allk; where, according to (2.19),

Z

Ω^{−}_{m}

g(x, u_{m}_{k}(x))u_{m}_{k}(x)dx

6C(1 +ku^{−}_{m}

kk), for allk. (2.48) Thus, combining (2.47) and (2.48),

Z

Ω

|∇u^{−}_{m}_{k}|^{2}dx−λ1

Z

Ω

(u^{−}_{m}_{k})^{2}dx

6C(1 +ku^{−}_{m}_{k}k), for allk. (2.49)
Next, divide on both sides of (2.49) byku^{−}_{m}_{k}k^{2}and use (2.26) to obtain

1−λ1

Z

Ω

(vm_{k})^{2}dx

6C 1

ku^{−}mkk^{2} + 1
ku^{−}mkk

, for allk. (2.50) It then follows from (2.23), (2.28) and (2.50) that

λ1

Z

Ω

(v)^{2}dx= 1,

from which we conclude thatvis a nontrivial solution of BVP (2.46). Consequently, sincevm60 for allm, according to (2.26), we obtain that

v=−ϕ1, (2.51)

where ϕ1 is the eigenfunction for the BVP (2.46) corresponding to the eigenvalue λ1with

ϕ_{1}>0 in Ω and kϕ_{1}k= 1.

We therefore obtain from (2.51) that

v <0 in Ω. (2.52)

Furthermore,

∂v

∂ν >0 on∂Ω, (2.53)

whereν denotes the outward unit normal vector to∂Ω. We can then conclude from (2.25), (2.29), in conjunction with (2.52) and (2.53), that

um_{k}(x)→ −∞ for a.e. x∈Ω. (2.54)
Thus, using (A5) and the Lebesgue dominated convergence theorem, we obtain
from (2.54) that

lim

k→∞

Z

Ω

G(x, u_{m}_{k}(x))dx=G_{−∞}|Ω|. (2.55)
It then follows from (2.55) and the first assertion in (2.1) that

lim

k→∞

1 2

Z

Ω

|∇umk|^{2}dx−λ_{1}
2

Z

Ω

(u^{−}_{m}

k)^{2}dx

=c+G_{−∞}|Ω|. (2.56)
Next, we go back to the estimate in (2.3) and setv=u^{+}_{m}

k to obtain

Z

Ω

|∇u^{+}_{m}_{k}|^{2}dx−
Z

Ω

g(x, u^{+}_{m}_{k})u^{+}_{m}_{k}dx

6εm_{k}ku^{+}_{m}_{k}k, for allk,

or, dividing byku^{+}_{m}_{k}k,

ku^{+}_{m}_{k}k −
Z

Ω

g(x, u^{+}_{m}_{k}) u^{+}_{m}_{k}
ku^{+}m_{k}kdx

6εm_{k}, for allk. (2.57)
Now, it follows from (2.54) that

u^{+}_{m}

k →0 a.e. ask→ ∞.

Therefore, it follows from the assumption thatg(x,0) = 0 in (A1), together with the Lebesgue dominated convergence theorem and the estimate in (2.57), that

k→∞lim ku^{+}_{m}

kk= 0. (2.58)

Next, setV = span{ϕ1}andW =V^{⊥}; so that,H_{0}^{1}(Ω) =V ⊕W.

Writeu^{−}_{m}_{k} =vk+wk, for eachk, wherevk ∈V andwk ∈W. Once again, use
the estimate in (2.3), this time withv=wk, to obtain

Z

Ω

|∇wk|^{2}dx−λ1

Z

Ω

w^{2}_{k}dx−
Z

Ω

g(x, um_{k}(x))wkdx

6εm_{k}kwkk, (2.59)
for allk.

Now, sincewk ∈W, we have that λ2

Z

Ω

w_{k}^{2}dx6
Z

Ω

|∇wk|^{2}dx, for allk, (2.60)
whereλ2denotes the second eigenvalue of theN-dimensional Laplacian over Ω with
Dirichlet boundary conditions. Consequently,

1−λ_{1}
λ2

kw_{k}k^{2}6
Z

Ω

|∇w_{k}|^{2}dx−λ_{1}
Z

Ω

w^{2}_{k}dx, for allk. (2.61)
Thus, settingα= 1−^{λ}_{λ}^{1}

2 in (2.61), we obtain from (2.61) and (2.59) that
αkw_{k}k^{2}6ε_{m}_{k}kw_{k}k+

Z

Ω

g(x, u_{m}_{k}(x))w_{k}dx

, for allk, (2.62) whereα >0.

Next, we divide on both sides of (2.62) bykwkkto get
αkwkk6εm_{k}+

Z

Ω

g(x, um_{k}(x)) wk

kw_{k}kdx

, for allk. (2.63) Now, it follows from (2.63), (2.54), assumption (A2), and the Lebesgue dominated convergence theorem that

lim

k→∞kw_{k}k= 0. (2.64)

Next, we observe that Z

Ω

|∇um_{k}|^{2}dx=
Z

Ω

|∇u^{+}_{m}_{k}|^{2}dx+
Z

Ω

|∇vk|^{2}dx+
Z

Ω

|∇wk|^{2}dx, for allk,

and Z

Ω

(u^{−}_{m}_{k})^{2}dx=
Z

Ω

v^{2}_{k}dx+
Z

Ω

w^{2}_{k}dx, for allk;

consequently, 1 2

Z

Ω

|∇um_{k}|^{2}dx−λ1

2 Z

Ω

(u^{−}_{m}_{k})^{2}dx

= 1

2ku^{+}_{m}_{k}k^{2}+1
2

Z

Ω

|∇wk|^{2}dx−λ1

2 Z

Ω

w^{2}_{k}dx,

(2.65)

for allk, where we have used the fact thatvk∈V for allk. It follows from (2.58), (2.64), (2.60) and (2.65) that

lim

k→∞

1 2

Z

Ω

|∇umk|^{2}dx−λ_{1}
2

Z

Ω

(u^{−}_{m}

k)^{2}dx

= 0. (2.66)

Combining (2.56) and (2.66) we obtain

G_{−∞}|Ω|+c= 0,

which is in direct contradiction with (1.9). We therefore conclude that (u^{−}_{m}) is
bounded.

Since, (u^{−}_{m}) is bounded, it follows from (2.22) that (u^{+}_{m}) is also bounded. Con-
sequently, (um) is bounded.

We will next proceed to show that (um) has a subsequence that converges
strongly inH_{0}^{1}(Ω). To see why this is the case, first write the functionalJ:H_{0}^{1}(Ω)→
Rdefined in (1.5) in the form

J(u) = 1 2

Z

Ω

|∇u|^{2}dx−
Z

Ω

Q(x, u(x))dx, for allu∈H_{0}^{1}(Ω),
where

Q(x, s) = Z s

0

q(x, ξ)dξ, for allx∈Ω ands∈R,

where q is as given in (1.3). It follows from (1.3) and the assumptions in (A2)
and (A3), thatq(x, s) has subcritical growth ins, uniformly inx∈Ω; so that, the
derivative map ofJ, ∇J: H_{0}^{1}(Ω)→H_{0}^{1}(Ω), is of the form

∇J =I− ∇Q, (2.67)

where∇Q:H_{0}^{1}(Ω)→H_{0}^{1}(Ω), given by
h∇Q(u), vi=

Z

Ω

g(x, u(x))v(x)dx, foru, v∈H_{0}^{1}(Ω),
is a compact operator.

Now, it follows from the second condition in (2.1) and (2.67) that

um− ∇Q(um)→0, as m→ ∞. (2.68)
Since we have already seen that the (PS)_{c} sequence (um) is bounded, we can
extract a subsequence, (um_{k}), of (um) that converges weakly to someu∈H_{0}^{1}(Ω).

Therefore, given that the map∇Q:H_{0}^{1}(Ω)→H_{0}^{1}(Ω) is compact, we have that

k→∞lim ∇Q(um_{k}) =∇Q(u). (2.69)
Thus, combining (2.68) and (2.69), we obtain that

k→∞lim um_{k}=∇Q(u).

We have therefore shown that (um) has a subsequence that converges strongly in
H_{0}^{1}(Ω), and the proof of the fact that J satisfies that (PS)_{c} condition, provided

thatc6=−G−∞|Ω|, is now complete.

3. Existence of a local minimizer

Assume thatgandGsatisfy conditions (A1)–(A7) hold. In this section, we will
use Ekeland’s Variational Principle and a cutoff technique similar to that used by
Chang, Li and Liu in [5] to prove the existence of a nontrivial solution of problem
(1.1) for the case in whichG_{−∞}60 in (A5).

To do that, we first defineeg∈C(Ω×R,R) by eg(x, s) =

(g(x, s), fors <0,

0, fors≥0. (3.1)

Define a corresponding functionalJe:H_{0}^{1}(Ω)→Rby
Je(u) = 1

2 Z

Ω

|∇u|^{2}dx−λ1

2 Z

Ω

(u^{−})^{2}dx−
Z

Ω

G(x, u)e dx, u∈H_{0}^{1}(Ω), (3.2)
where

G(x, s) =e Z s

0 eg(x, ξ)dξ, forx∈Ω ands∈R. (3.3)
Then,Je∈C^{1}(H_{0}^{1}(Ω),R). We claim thatJeis bounded below. In fact, by condition
(A5) and (3.1), it follows that

|G(x, s)|e 6Mo, for allx∈Ω ands∈R, (3.4)
for someM_{o}>0. Then, using (3.1) and (3.4), we can write

Je(u) = 1 2 Z

Ω

|∇u|^{2}dx−λ1

2 Z

Ω

(u^{−})^{2}dx−
Z

Ω

G(x, u)dx,e

> 1

2ku^{+}k^{2}+1

2ku^{−}k^{2}−λ1

2 ku^{−}k_{L}2−Mo|Ω|,

(3.5)

for allu∈H_{0}^{1}(Ω). It then follows from (3.5) and the Poincar´e inequality that
J(u)e >−Mo|Ω|, for allu∈H_{0}^{1}(Ω);

so that Jeis bounded below. Thus, the infimum of Jeover H_{0}^{1}(Ω) exists; we can,
therefore define

c1= inf

u∈H_{0}^{1}(Ω)

Je(u). (3.6)

Notice that, sinceJe(0) = 0, we must havec_{1}60. In fact, we presently show that,
if (A6) and (A7) hold, then

c_{1}<0. (3.7)

To do this, first use (1.3) and (A7) to compute lim

s→0^{−}

g(x, s)

s =a−λ_{1};
so that

lim

s→0^{−}

g(x, s) s >0,

for all x∈Ω, by the assumption ona in (A7). Consequently, there exists s_{1} <0
such that

g(x, s)<0, fors1< s <0,

and allx∈Ω. It then follows from the definition ofGe in (3.3) that

G(x, s)e >0 fors_{1}< s <0, and allx∈Ω. (3.8)

Next, letε >0 be small enough so that

s_{1}<−εϕ_{1}(x)<0, for allx∈Ω. (3.9)
We then have that

Z

Ω

G(x,e −εϕ(x))dx >0, (3.10) by (3.8) and (3.9). It then follows from the definition ofJein (3.2) and (3.10) that

Je(−εϕ1) =− Z

Ω

G(x,e −εϕ(x))dx <0.

Consequently, in view of the definition ofc1 in (3.6), we obtain thatc1<0, which is (3.7).

We now use (3.6) and a consequence of Ekeland’s Variational Principle (see [10,
Theorem 4.4]) to obtain, for each positive integerm, um∈H_{0}^{1}(Ω) such that

Je(u_{m})6 inf

u∈H_{0}^{1}(Ω)

Je(u) + 1

m, for allm, (3.11)

and

kJe^{0}(um)k ≤ 1

m, for allm;

we therefore obtain a (PS)_{c} sequence for c = c1. Consequently, if Jehappens to
satisfy the (PS)_{c} condition atc=c1, we would conclude thatc1is a critical value
of Je. We will show shortly that this is the case if we assume thatG_{−∞} given in
(A5) satisfies

G_{−∞}60. (3.12)

We will first establish thatJesatisfies the (P S)c provided thatc6=−G−∞|Ω|.

Proposition 3.1. Assume that g and Gsatisfy (A1), (A2) and(A5), and define
Jeas in (3.2), where Ge is given in (3.3) and (3.1). Then, Jesatisfies the (P S)_{c}
condition forc6=−G_{−∞}|Ω|.

Proof. Assume thatc6=−G−∞|Ω|and let (um) be a (P S)c sequence forJe; that is, 1

2 Z

Ω

|∇um|^{2}dx−λ1

2 Z

Ω

(u^{−}_{m})^{2}dx−
Z

Ω

G(x, ue m)dx→c, asm→+∞, (3.13) and,

Z

Ω

∇u_{m}· ∇ϕ dx+λ_{1}
Z

Ω

u^{−}_{m}ϕ dx−
Z

Ωeg(x, u_{m})ϕ dx

≤ε_{m}kϕk, (3.14)
for all m and all ϕ∈ H_{0}^{1}(Ω), where (ε_{m}) is a sequence of positive numbers such
thatεm→0 asm→ ∞. Write um=u^{+}_{m}−u^{−}_{m}. We will show that (u^{+}_{m}) and (u^{−}_{m})
are bounded sequences.

First, let’s see that (u^{+}_{m}) is bounded. Setting ϕ=u^{+}_{m}in (3.14) we have

Z

Ω

|∇u^{+}_{m}|^{2}dx−
Z

Ωeg(x, um)u^{+}_{m}dx

≤εmku^{+}_{m}k for allm. (3.15)
By (3.1) and the assumption in (A2), it can be shown thateg(x, um) is bounded for
allx∈Ω. Then, using H¨older and Poincar´e’s inequalities, we obtain that

Z

Ωeg(x, um)u^{+}_{m}dx

≤Cku^{+}_{m}k, (3.16)

for some constantC >0. Then, from (3.15) and (3.16), we obtain that
ku^{+}_{m}k^{2}≤(C+εm)ku^{+}_{m}k, for allm,

which shows that (u^{+}_{m}) is a bounded sequence.

Next, let us show that (u^{−}_{m}) is a bounded sequence. Suppose that this is not the
case; then, passing to a subsequence if necessary, we may assume that

ku^{−}_{m}k → ∞ asm→ ∞. (3.17)
Define

v_{m}=− u^{−}_{m}

ku^{−}mk, for allm. (3.18)

Then, sincekv_{m}k= 1 for all m, passing to a further subsequences if necessary, we
may assume that there isv∈H_{0}^{1}(Ω) such that

vm* v (weakly) in H_{0}^{1}(Ω), asm→ ∞; (3.19)

vm→v in L^{2}(Ω), asm→ ∞; (3.20)

vm(x)→v(x) for a.e. xin Ω, asm→ ∞. (3.21)
Now, writingum=u^{+}_{m}−u^{−}_{m} in (3.14) we have

Z

Ω

∇u^{+}_{m}· ∇ϕ dx−
Z

Ω

∇u^{−}_{m}· ∇ϕ dx+λ1

Z

Ω

u^{−}_{m}ϕ dx−
Z

Ωeg(x, um)ϕ dx

6εmkϕk,
for allϕ∈H_{0}^{1}(Ω) and allm, from which we obtain that

−

Z

Ω

∇u^{−}_{m}· ∇ϕ dx+λ_{1}
Z

Ω

u^{−}_{m}ϕ dx−

Z

Ωeg(x, u_{m})ϕ dx

6(ε_{m}+Cku^{+}_{m}k)kϕk, (3.22)
for all ϕ∈ H_{0}^{1}(Ω), all m, and some constant C >0, by the Cauchy-Schwarz and
Poincar´e inequalities.

Now, we divide both sides of (3.22) byku^{−}_{m}k and use (3.18) to obtain

Z

Ω

∇v_{m}·∇ϕ dx−λ_{1}
Z

Ω

v_{m}ϕ dx−

Z

Ω

eg(x, u_{m})
ku^{−}mk ϕ dx

6ε_{m}+Cku^{+}_{m}k
ku^{−}mk

kϕk, (3.23)
for allϕ∈H_{0}^{1}(Ω) and allm. Sinceeg is bounded, by condition (A2) and (3.1), we
obtain from (3.17) that

m→+∞lim

eg(x, um(x))

ku^{−}mk = 0, for a. e. x∈Ω.

It then follows from the Lebesgue dominated convergence theorem that

m→+∞lim Z

Ω

eg(x, u_{m}(x))

ku^{−}mk ϕ dx= 0, for allϕ∈H_{0}^{1}(Ω). (3.24)
Therefore, using (3.19), (3.20), (3.24), (3.17), the fact that the sequence (u^{+}_{m}) is
bounded, and lettingm→ ∞in (3.23), we obtain

Z

Ω

∇v· ∇ϕ dx−λ1

Z

Ω

vϕ dx= 0, for allϕ∈H_{0}^{1}(Ω);

so that,v is a weak solution of the BVP

−∆u=λ1u, in Ω;

u= 0, on∂Ω. (3.25)

Next, we setϕ=vmin (3.23) to get

1−λ1

Z

Ω

v^{2}_{m}dx−
Z

Ω

eg(x, u_{m})
ku^{−}mk vmdx

6ε_{m}+Cku^{+}_{m}k

ku^{−}mk , for allm, (3.26)
where we have also used the definition ofvm in (3.18).

Now, using the Cauchy-Schwarz and Poincar´e inequalities, we obtain that

Z

Ω

eg(x, u_{m})
ku^{−}mk v_{m}dx

6 C

ku^{−}mk, for allm, (3.27)
for some positive constant C, since eg is bounded. We then get from (3.27) and
(3.17) that

m→∞lim Z

Ω

eg(x, u_{m})

ku^{−}mk vmdx= 0. (3.28)

Thus, lettingm→ ∞in (3.26) and using (3.20), (3.28), and (3.17), we obtain that λ1

Z

Ω

v^{2}dx= 1,
which shows thatv is a nontrivial solution of (3.25).

Now, it follows from (3.18) and (3.21) that

v(x)60, for a. e. x∈Ω.

Consequently, sincev is nontrivial, it must be the case that

v=−ϕ1, (3.29)

where ϕ1 is the eigenfunction of the BVP problem (3.25) corresponding to the eigenvalue λ1 with ϕ1 > 0, kϕ1k = 1. Thus, v < 0 ∈ Ω and ∂v/∂ν > 0 on ∂Ω, whereν is the outward unit normal vector to∂Ω.

Next, we writeu_{m}=u^{+}_{m}−u^{−}_{m} and use (3.18) to get
u_{m}

ku^{−}mk = u^{+}_{m}

ku^{−}mk +v_{m}, for allm;

so that, by the fact that (u^{+}_{m}) is bounded and (3.17), we may assume, passing to a
further subsequence if necessary, that

um(x)

ku^{−}mk → −ϕ1(x), for a. e. x∈Ω, as m→ ∞, (3.30)
where we have also used (3.21) and (3.29). It then follows from (3.30) that

um(x)→ −∞ for a. e. x∈Ω, as m→ ∞. (3.31) Then, using condition (A5) and the Lebesgue dominated convergence theorem, we conclude from (3.13) that

m→∞lim 1

2 Z

Ω

|∇um|^{2}dx−λ1

2 Z

Ω

(u^{−}_{m})^{2}dx

=c+G_{−∞}|Ω|. (3.32)
Next, we divide both sides of (3.15) byku^{+}_{m}kto obtain

ku^{+}_{m}k −

Z

Ω

eg(x, um)
ku^{+}mk u^{+}_{m}dx

6εm, for allm. (3.33) Notice that

Z

Ω

g(x, ue _{m})

ku^{+}mk u^{+}_{m}dx=
Z

Ωeg(x, u^{+}_{m}) u^{+}_{m}

ku^{+}mkdx, for allm;

so that, using the Cauchy-Schwarz and Poincar´e inequalities,

Z

Ω

eg(x, um)
ku^{+}mk u^{+}_{m}dx

6C sZ

Ωeg(x, u^{+}m(x))^{2}dx, for allm, (3.34)
and some positive constantC. Now, it follows from (3.31) that

u^{+}_{m}(x)→0 for a.ex∈Ω, asm→ ∞;

consequently, using the assumption (A1) along with the Lebesgue dominated con- vergence theorem, we obtain from (3.34) that

m→∞lim Z

Ω

g(x, ue _{m})

ku^{+}mk u^{+}_{m}dx= 0. (3.35)
Therefore, lettingmtend to ∞in (3.33) and using (3.35) we obtain that

m→∞lim ku^{+}_{m}k= 0. (3.36)

Thus, combining (3.32) and (3.36) we can then write

m→∞lim 1

2 Z

Ω

|∇u^{−}_{m}|^{2}dx−λ_{1}
2

Z

Ω

(u^{−}_{m})^{2}dx

=c+G−∞|Ω|. (3.37) We may now proceed as in the proof of Proposition 2.1 in Section 2 to show that

m→∞lim 1

2 Z

Ω

|∇u^{−}_{m}|^{2}dx−λ1

2 Z

Ω

(u^{−}_{m})^{2}dx

= 0.

Hence, in view of (3.37), we obtain thatc+G_{−∞}|Ω|= 0, which contradicts the as-
sumption thatc6=−G_{−∞}|Ω|. We therefore conclude that (u^{−}_{m}) must be a bounded
sequence. Thus, since we have already seen that (u^{+}_{m}) is bounded, we see that (um)
is bounded.

We have therefore shown that any (PS)_{c} sequence with c6=−G_{−∞}|Ω|must be
bounded. The remainder of this proof now proceeds as in the proof of Proposition
2.1 presented in Section 2, using in this case the fact thateg is bounded.

Now, if we assume that the value G−∞ given in (A5) satisfies the condition in
(3.12), then we would have that−G−∞|Ω|>0. Consequently, in view of (3.7), we
see that the value ofc_{1} given in (3.6) is such that

c_{1}<−G_{−∞}|Ω|;

therefore, Je satisfies the (PS)_{c} condition at c = c1. Hence, by the discussion
preceding the statement of Proposition 3.1,c1 is a critical value ofJe. Thus, there
exists u1 ∈H_{0}^{1}(Ω) that is a global minimizer forJe. We note that u1 6≡0 in Ω by
(3.7).

Now, since the functionegdefined in (3.1) is locally Lipschitz (refer to assumption
in (A6)), it follows thatu_{1}is a classical solution of the problem

−∆u=−λ_{1}u^{−}+eg(x, u), in Ω;

u= 0, on∂Ω, (3.38)

(see Agmon [1]).

Let Ω_{+} ={x∈Ω|u_{1}(x)>0}. Then, by the definition of eg in (3.1), u_{1} solves
the BVP

−∆u= 0, in Ω+;

u= 0, on∂Ω, (3.39)

which has only the trivial solutionu≡0. Thus, Ω+=∅and thereforeu160 in Ω.

Before we state the main result of this section, though, we will discuss some properties of the critical pointu1.

Since we have already seen thatu_{1}60 in Ω, it follows from the definition ofegin
(3.1) thatu_{1} is also a solution of the BVP (1.1); consequently,u_{1} is also a critical
point ofJ. We will show shortly thatu_{1} is a local minimizer forJ.

Sinceu_{1}is a solution of the BVP (3.38), thenu_{1} is also a solution of the BVP

−∆u−p(x)u=λ1u1−g^{−}(x, u1(x)), in Ω;

u= 0, on∂Ω, (3.40)

where

p(x) =

(_{g}+(x,u_{1}(x))

u1(x) , ifu1(x)<0;

0, ifu_{1}(x) = 0. (3.41)

Now, it follows from (3.40) and the fact thatu_{1}(x)60 for allx∈Ω thatu_{1}solves

−∆u−p(x)u60, in Ω;

u= 0, on∂Ω. (3.42)

Thus, since p(x) 6 0, according to (3.41), we can apply the Hopf’s Maximum Principle (see, for instance, [11, Theorem 4, p. 333]) to conclude that

u_{1}(x)<0, for allx∈Ω, (3.43)
sinceu1 is nontrivial, and

∂u1

∂ν (x)>0, forx∈∂Ω, (3.44)

where ν denotes the outward unit normal vector to the surface∂Ω. We can then
use (3.43) and (3.44), and the assumption that Ω is bounded to show that there
existsδ >0 such that, if u∈C^{1}(Ω)∩H_{0}^{1}(Ω) is such that

ku−u1k_{C}1(Ω)< δ,
then

u(x)<0, for allx∈Ω.

Consequently, ifuis in aδ-neighborhood ofu1 in theC^{1}(Ω) topology, then
J(u) =Je(u)>Je(u_{1}) =J(u_{1});

so thatu1 is a local minimizer ofJ in the C^{1}(Ω) topology. It then follows from a
result of Brezis and Nirenberg in [4] that u1 is also a local minimizer for J in the
H_{0}^{1}(Ω) topology. We have therefore demonstrated the following theorem.

Theorem 3.2. Assume that g and Gsatisfy conditions (A1)–(A4). Assume also
that (A6) and (A7) are satisfied. If (A5) holds true with G−∞ 6 0, then the
BVP (1.1)has a nontrivial solution, u1, that is a local minimizer of the functional
J: H_{0}^{1}(Ω)→Rdefined in (1.5).

In the next section, we will provide additional conditions on the nonlinearity,g, that will allow us to show that the functionalJ defined in (1.5) has another local minimizer.