The problem under consideration is as follows: Find u ∈ W01,p(Ω) such that  −∆pu(x

Hemivariational inequalities governed by the p-Laplacian - Dirichlet problem

by

Zdzis law Naniewicz

Cardinal Stefan Wyszy´nski University Department of Mathematics and Natural Sciences.

College of Science

Dewajtis 5, 01-815 Warsaw, Poland

e-mail: naniewicz@uksw.edu.pl; naniewicz.z@acn.waw.pl Abstract: A hemivariational inequality involving p-Laplacian is studied under the hypothesis that the nonlinear part fulfills the uni- lateral growth condition (Naniewicz, 1994). The existence of solu- tions for problems with Dirichlet boundary conditions is established by making use of Chang’s version of the critical point theory for non- smooth locally Lipschitz functionals (Chang, 1981), combined with the Galerkin method. A class of problems with nonlinear poten- tials fulfilling the classical growth hypothesis without Ambrosetti- Rabinowitz type assumption is discussed. The approach is based on the recession technique introduced in Naniewicz (2003).

Keywords: Dirichlet problem, hemivariational inequality, uni- lateral growth condition, critical point theory, locally Lipschitz func- tional.

1. Introduction

Let Ω ⊆ R^N be a bounded domain with Lipschitz boundary ∂Ω. The problem under consideration is as follows: Find u ∈ W₀^1,p(Ω) such that

 −∆pu(x) ∈ −∂j(x, u(x)) a.e. on Ω

u|∂Ω = 0, 2 ≤ p < ∞, (1)

where −∆pu := − div |Du|^p−2Du

stands for the p-Laplacian operator. By

∂j(x, u) we denote the generalized gradient of Clarke (Clarke, 1983) of a locally Lipschitz R ∋ ξ 7→ j(x, ξ) (for a.e. x ∈ Ω). For the right hand side of (1) we suppose that it satisfies the unilateral growth condition of the form (Naniewicz, 1994):

j⁰(x; ξ, −ξ) ≤ κ(1 + |ξ|^q), ∀ ξ ∈ R, for a.e. x ∈ Ω, q < p^⋆.

Thus the problem to be studied involves nonlinear, nonconvex function j(·, u) which is not summable for every u ∈ W₀^1,p(Ω) and consequently, the correspond- ing energy functional R(u) = ¹_pkDuk^p_Lp(Ω;R^N)+R

Ωj(x, u(x)) dx is not locally Lipschitz and has no longer the whole space W₀^1,p(Ω) as its effective domain. The direct use of the critical point theory developed for locally Lipschitz functionals (Chang, 1981) is therefore not available. We use the Galerkin method and solve the discretized problems in finite dimensional subspaces of W₀^1,p(Ω) ∩ L^∞(Ω) by making use of the recession technique for semicoercive problems introduced in Naniewicz (2003) and then pass to the limit to get a solution.

The class of hemivariational inequalities considered in the paper can be re- ferred to as variational problems with discontinuities, widely studied recently.

For elliptic problems with the classical growth conditions we refer to Montre- anu and Panagiotopoulos (1995, 1996, 1999), Goeleven, Motreanu and Pana- giotopoulos (1997), Radulescu (1993), Gasi´nski and Papageorgiou (2001b). Non- smooth problems within the framework of the unilateral growth conditions can be found in Montreanu and Naniewicz (2001, 2002, 2003), Halidias and Naniewicz (2004), Naniewicz (2003) and the references quoted there. For elliptic problems involving p-Laplacian we refer to Arcoya and Orsina (1997), Bouchala and Drabek (200), Anane and Gossez (1990) (smooth potentials) and to Gasi´nski and Papageorgiou (2001a, c), Papalini (2002), Halidias and Naniewicz (2004) in the case of nonsmooth potentials.

The notion of hemivariational inequalities has been first introduced in the early eighties with the works of P. D. Panagiotopoulos (Panagiotopoulos, 1981, 1983). The main reason for its birth was the need for description of important problems in physics and engineering, where nonmonotone, multivalued bound- ary or interface conditions occur, or where some nonmonotone, multivalued relations between stress and strain, or reaction and displacement have to be taken into account. The theory of hemivariational inequalities (as the gen- eralization of variational inequalities, see Duvaut and Lions (1972) has been proved to be very useful in the understanding of many problems of mechanics and engineering involving nonconvex, nonsmooth energy functionals. For the general study of hemivariational inequalities in both scalar and vector-valued function spaces the reader is referred to Panagiotopoulos (1985, 1993), Montre- anu and Panagiotopoulos (1999), Montreanu and Naniewicz (1996), Naniewicz and Panagiotopoulos (1995) and the references quoted there.

2. Mathematical background

Let us recall some facts and definitions from the critical point theory for locally Lipschitz functionals and the generalized gradient of Clarke (Clarke, 1983).

Let Y be a subset of a Banach space X. A function f : Y → R is said to satisfy a Lipschitz condition (on Y ) provided that, for some nonnegative

scalar K, one has

|f (y) − f (x)| ≤ Kky − xkX

for all points x, y ∈ Y . Let f be Lipschitz near a given point x, and let v be any vector in X. The generalized directional derivative of f at x in the direction v, denoted by f⁰(x; v), is defined as follows:

f⁰(x; v) = lim sup

y→x t↓0

f (y + tv) − f (y) t

where y is a vector in X and t a positive scalar. If f is Lipschitz of rank K near x then the function v → f⁰(x; v) is finite, positively homogeneous, subadditive and satisfies the conditions |f⁰(x; v)| ≤ KkvkX and f⁰(x; −v) = (−f )⁰(x; v).

Now we are ready to introduce the generalized gradient ∂f (x) defined in Clarke (1983):

∂f (x) = {w ∈ X^⋆: f⁰(x; v) ≥ w, v

X for all v ∈ X}.

Some basic properties of the generalized gradient of locally Lipschitz functionals are the following:

(a) ∂f (x) is a nonempty, convex, weakly-star compact subset of X^⋆ and kwkX^⋆ ≤ K for every w in ∂f (x);

(b) For every v in X, one has f⁰(x; v) = max{

w, v

: w ∈ ∂f (x)};

(c) If f1, f2are locally Lipschitz functions then

∂(f1+ f2) ⊆ ∂f1+ ∂f2.

Let us recall the (P.S.)-condition introduced by Chang (Chang, 1981).

Definition A locally Lipschitz function f is said to satisfy the Palais - Smale condition if any sequence {xn} along which |f (xn)| is bounded and

λ(xn) = min

w∈∂f (xn)kwkX^⋆ → 0, possesses a convergent subsequence.

Let us mention some facts about the first eigenvalue of the p-Laplacian.

Consider the first nonzero eigenvalue λ1of (−∆p, W₀^1,p(Ω)). It is well known (see Lindqvist, 1990) that λ1> 0 and it is characterized by the Rayleigh quo- tient:

λ1:= inf

(kDwk^p_Lp(Ω;R^N)

kwk^p_Lp(Ω)

: w ∈ W₀^1,p(Ω), w 6= 0 )

.

Each eigenfunction w ∈ W₀^1,p(Ω) corresponding to λ1 has the properties that kDwk^p_Lp(Ω;R^N)= λ1kwk^p_Lp(Ω) and it is a solution of the problem

( −∆pw = λ1|w|^p−2w a.e. on Ω

w|∂Ω = 0, 2 ≤ p < ∞. (2)

Moreover, the generalized Poincar´e inequality due to Fleckinger-Pell´e-Tk´aˇc (2002) holds: There exists a positive constant c > 0 such that:

Z

Ω

|Du|^pdx − λ1

Z

Ω

|u|^pdx ≥ c



|e|^p−2 Z

Ω

|Dθ|^p−2|Dbu|²dx + Z

Ω|Dbu|^pdx

 ,

∀ u ∈ W₀^1,p(Ω), (3) where θ is the λ1-eigenfunction and u = eθ + bu is an orthogonal decomposition of u in L²(Ω), e = kθk⁻²_L2(Ω)

u, θ

L²(Ω) and b u, θ

L²(Ω)= 0.

Let f : X → R be a locally Lipschitz function on a Banach space. A point x ∈ X is said to be a critical point of f if 0 ∈ ∂f (x) and c = f (x) is then its critical value.

The theorems below characterize conditions under which the existence of critical points follows. They are due to Chang (Chang, 1981) and extend to a nonsmooth setting the well known classical results of the critical point theory.

Theorem 2.1 If a locally Lipschitz function f : X → R on the reflexive Banach space X satisfies the (PS)-condition and there exist a positive constant ρ > 0 and e ∈ X with kek > ρ such that

max{f (0), f (e)} < inf

kxk=ρ{f (x)},

then f has a critical point u ∈ X with its critical value c = f (u) characterized by

c = inf

g∈G max

t∈[0,1]f (g(t)) where

G = {g ∈ C([0, 1], X) : g(0) = 0, g(1) = e}.

Theorem 2.2 Suppose that a locally Lipschitz function f : X → R satisfies the (PS)-condition and is bounded from below. Then c = infX{f (x)} is a critical value of f .

3. Auxiliary results

Let us denote by V0 = {sθ}_s∈R the one-dimensional eigenspace spanned by the eigenfunction θ corresponding to the first eigenvalue λ1 of −∆p, W₀^1,p(Ω)

.

It is well known (see Anane, 1988, Lindqvist, 1990) that θ ∈ L^∞(Ω), θ does not change its sign in Ω. Therefore one can normalize the eigenfunction by assuming that θ > 0 a.e. in Ω and kθk_W^1,p

0 (Ω) = 1. By V^⊥ we denote the orthogonal complement of V0 in L²(Ω). Accordingly, for any u ∈ W₀^1,p(Ω) the decomposition follows

u = eθ + bu with e ≥ 0, θ ∈ {±θ} ⊂ V0, bu ∈ bV , (4) where bV := V^⊥∩ W₀^1,p(Ω), and by (3) we have the inequality

Z

Ω

|Du|^pdx − λ1

Z

Ω

|u|^pdx ≥ c Z

Ω|Dbu|^pdx, ∀ u ∈ W₀^1,p(Ω). (5) Lemma 3.1 Assume that

(H1) j(·, 0) ∈ L¹(Ω) and j(x, ·) is Lipschitz continuous on the bounded subsets of R uniformly with respect to x ∈ Ω, i.e., ∀ r > 0 ∃ Kr > 0 such that

∀ |y1|, |y2| ≤ r,

|j(x, y1) − j(x, y2| ≤ Kr|y1− y2|, for a.e. x ∈ Ω;

(H2) One of the two conditions below holds (the Ambrosetti-Rabinowitz type conditions):

(i) There exist µ > p, 1 ≤ σ < p, a ∈ L¹(Ω) and a constant k ≥ 0 such that µj(x, ξ)−j⁰(x, ξ; ξ)+λ1µ−p

p |ξ|^p≥ −a(x)−k|ξ|^σ, ∀ ξ ∈ R and for a.e. x ∈ Ω;

(ii) There exist 0 < ν < p, 1 ≤ σ < p, a ∈ L¹(Ω) and a constant k ≥ 0 such that

−νj(x, ξ)−j⁰(x, ξ; −ξ)+λ1p−ν

p |ξ|^p≥ −a(x)−k|ξ|^σ, ∀ ξ ∈ R and for a.e. x ∈ Ω;

(H3) Suppose that J^∞(θ) + λ1

Z

Ω

|θ|^pdx > 0 for each θ ∈ {±θ},

where

J^∞(θ) := lim inf

t→+∞

η−→θ

Lp(Ω)

1 t^p−1

Z

Ω

−j⁰ x, tη(x); −θ(x)

dx, θ ∈ {±θ},

is the recession function of nonconvex, nonsmooth J(·) =R

Ωj(x, ·) dx (see Naniewicz, 2003, and also Goeleven and Th´era, 1995, Baiocchi, Buttazzo, Gastaldi and Tomarelli, 1988).

Moreover, suppose that for a sequence {un} ⊂ W₀^1,p(Ω) ∩ L^∞(Ω) there exists εnց 0 such that the conditions below are fulfilled:

Z

Ω

|Dun(x)|^p−2

Dun(x), Dv(x) − Dun(x)

R^Ndx +

Z

Ω

j⁰ x, un(x); v(x) − un(x)

dx ≥ −εnkv − unk_W^1,p

0 (Ω),

∀ v ∈ Lin({un, θ}), (6) and 

1 p

Z

Ω

|Dun(x)|^pdx + Z

Ω

j x, un(x)
dx

 ≤ C, C > 0. (7)

where Lin({un, θ}) is the linear subspace of W₀^1,p(Ω) spanned by {θ, un}. Then the sequence {un} is bounded in W₀^1,p(Ω), i.e. there exists M > 0 such that

kunk_W^1,p

0 (Ω)≤ M. (8)

Proof. Suppose on the contrary that the claim is not true, i.e. there exists a sequence {un}^∞_n=1 ⊂ W₀^1,p(Ω) ∩ L^∞(Ω) with kunk_W^1,p

0 (Ω) → ∞ for which (6) and (7) hold. Under (H2)(i) combining (7) multiplied by µ > p with (6) (with v = 2un substituted) yields

µC + εnkunk_W^1,p

0 (Ω)≥^µ−p_p 

kDunk^p_Lp(Ω;R^N)− λ1kunk^p_Lp(Ω)



+ Z

Ω

µj(un) − j⁰(un; un) + λ1µ−p p |un|^p

dx. (9)

Taking into account the decomposition un= enθn+ bun, where bun∈ bV , en≥ 0, θn∈ {±θ}, kθk_W^1,p

0 (Ω)= 1, by (5) and (H2)(i) we have µC +εnkunk_W^1,p

0 (Ω)≥ c^µ−p_p kD(bun)k^p_Lp(Ω;R^N)−c1kbun+enθnk^σ_Lp(Ω)−kakL¹(Ω). (10) Hence

µC + εn(kbunk_W^1,p

0 (Ω)+ en) ≥ c^µ−p_p kDbunk^p_Lp(Ω;R^N)

−c2kbunk^σ_W1,p

0 (Ω)− c3e^σ_n− kak_L¹(Ω). (11) Thus, it follows that en→ ∞ because otherwise we would get the boundedness of {bun} and consequently the boundedness of {un} in W₀^1,p(Ω), contrary to our supposition. Dividing this inequality by en yields

µC

en + εn(k^ub_enⁿk_W^1,p

0 (Ω)+ 1) ≥ e^p−1_n c^µ−p_p kD(^ub_eⁿ_n)k^p_Lp(Ω;R^N)

−e^σ−1_n c2k^bu_eⁿ

nk^σ_W1,p

0 (Ω)− e^σ−1_n c3−^kak_e^L1(Ω)

n . (12)

which means that {^ub_eⁿ

n} is bounded in W₀^1,p(Ω). Further, in view of σ < p and en→ ∞, this leads to the conclusion that

k^bu_eⁿ

nk_W^1,p

0 (Ω)→ 0. (13)

Now let us turn back to (6). By passing to a subsequence one can suppose also that θn= θ (or θn= −θ). Thus, substituting v = bun into (6) yields

e^p_n Z

Ω

|D(^ub_eⁿ

n) + Dθ)|^p−2 D(^ub_eⁿ

n) + Dθ), −Dθ

R^Ndx +en

Z

Ω

j⁰ en(^bu_eⁿ

n + θ); −θ

dx ≥ −εnen. Hence

εn

e^p−1n ≥ Z

Ω

h− ¹

e^p−1n j⁰ en(^ub_eⁿ

n + θ); −θ

− λ1|^ub_eⁿ

n + θ|^p−2(^ub_eⁿ

n + θ)(−θ)i dx +

Z

Ω

|D(^bu_eⁿ

n) + Dθ)|^p−2 D(^bu_eⁿ

n) + Dθ), Dθ

R^Ndx

−λ1

Z

Ω

|^ub_eⁿ

n + θ|^p−2(^ub_eⁿ

n + θ)θ dx.

Now we are ready to pass to the limit with n → ∞. For this purpose notice that in view of (13) there is

n→∞lim

Z

Ω

|D(^bu_eⁿ

n) + Dθ)|^p−2 D(^bu_eⁿ

n) + Dθ), Dθ

R^Ndx

−λ1

Z

Ω

|^ub_eⁿ

n + θ|^p−2(^ub_eⁿ

n + θ)θ dx



= kDθk^p_Lp(Ω;R^N)− λ1kθk^p_Lp(Ω)= 0.

Accordingly, we arrive at 0 ≥ lim sup

n→∞

1 e^p−1n

Z

Ω

−j⁰ en(^bu_enⁿ+θ); −θ
dx+λ1

Z

Ω

|θ|^pdx ≥ J^∞(θ)+λ1

Z

Ω

|θ|^pdx.

But this contradicts (H3).

Under (H2)(ii), combining (7) multiplied by ν < p with (6) (with v = 0 substituted) yields

νC + εnkunk_W^1,p

0 (Ω)≥ ^p−ν_p 

kDunk^p_Lp(Ω;R^N)− λ1kunk^p_Lp(Ω)



+ Z

Ω

−νj(un) − j⁰(un; −un) + λ1p−ν p |un|^p

dx. (14)

Now we can proceed as previously to establish the result. The proof of Lemma 3.1 is complete.

Lemma 3.2 Assume that (H1) and the hypotheses below hold:

(H4) The unilateral growth condition (Naniewicz, 1994): There exist p < q <

p^∗= _N−p^{N p} , and a constant κ ≥ 0 such that

j⁰(x, ξ; −ξ) ≤ κ(1 + |ξ|^q), ∀ ξ ∈ R and for a.e. x ∈ Ω;

(H5) Uniformly for a.e. x ∈ Ω,

lim inf

ξ→0

pj(x, ξ)

|ξ|^p ≥ φ(x) ≥ −λ1,

with φ(x) ∈ L^∞(Ω) and φ(x) > −λ1 on a set of positive measure.

Then there exists ρ > 0 such that R(u) :=¹_pkDuk^p_Lp(Ω;R^N)+

Z

Ω

j(u) dx ≥ η, η = const > 0, (15)

is valid for any u ∈ W₀^1,p(Ω) ∩ L^∞(Ω) with kuk_W^1,p

0 (Ω)= ρ.

Proof. Suppose the assertion is not true. Thus there exist sequences {un} ⊂ W₀^1,p(Ω) ∩ L^∞(Ω) and ρn ց 0 such that kunk_W^1,p

0 (Ω)= ρn and R(un) ≤ ρ_n^p+1. So we have

kDunk^p_Lp(Ω;R^N)+ Z

Ω

pj(un) dx ≤ pρ_n^p+1. (16)

Further, from (H5) it follows that for any ε > 0, uniformly for all x ∈ Ω one can find δ > 0 such that

pj(x, ξ) ≥ φ(x)|ξ|^p− ε|ξ|^p, |ξ| ≤ δ.

Moreover, (H4) allows to conclude that (see Lemma 2.1, pp. 119-120, Naniewicz, 1997):

j(x, ξ) ≥ −κ0(1 + |ξ|^q), ∀ ξ ∈ R, for a.e. x ∈ Ω, κ0= const > 0. (17) Thus it is easy to see that

pj(x, ξ) ≥ (φ(x) − ε)|ξ|^p− γ|ξ|^q, ∀ ξ ∈ R, (18) for some positive γ = γ(δ) > 0. Then by (16) it follows

kDunk^p_Lp(Ω;R^N)− λ1kunk^p_Lp(Ω)+ Z

Ω

h(φ(x) − ε)|un(x)|^p+ λ1|un(x)|^pi dx

≤ pρ_n^p+1+ γ Z

Ω

|un(x)|^qdx. (19)

Since W₀^1,p(Ω) is continuously embedded into L^q(Ω) we have kDunk^p_Lp(Ω;R^N)− λ1kunk^p_Lp(Ω)+

Z

Ω

(φ(x) + λ1− ε)|un(x)|^pdx

≤ pρ^p+1_n + γ1kunk^q

W₀^1,p(Ω), γ1= const > 0. (20) Dividing inequality (20) by ρ_n^pyields

kDynk^p_Lp(Ω;R^N)− λ1kynk^p_Lp(Ω)+ Z

Ω

(φ(x) + λ1− ε)|yn(x)|^pdx

≤ pρn+ γ1ρ^q−p_n . (21) The norms kD(·)k_L^p_(Ω;R^N₎ and k·k_W^1,p

0 (Ω) are equivalent on W₀^1,p(Ω) and kynk_W^1,p

0 (Ω) = 1. Therefore we can suppose that for a subsequence (again denoted by the same symbol) yn → y weakly in W₀^1,p(Ω) and yn → y strongly in L^p(Ω) (the Rellich theorem) for some y ∈ W₀^1,p(Ω). Passing to the limit and the weak lower semicontinuity of the norm allow the conclusion

kDyk^p_Lp(Ω;R^N)− λ1kyk^p_Lp(Ω)+ Z

Ω

(φ(x) + λ1− ε)|y(x)|^pdx ≤ 0, (22) which is valid for an arbitrary ε > 0. Therefore we get

kDyk^p_Lp(Ω;R^N)− λ1kyk^p_Lp(Ω)+ Z

Ω

(φ(x) + λ1)|y(x)|^pdx ≤ 0. (23) Application of the Rayleigh quotient characterization of λ1 and (H5) leads to the equalities

kDyk^p_Lp(Ω;R^N)= λ1kyk^p_Lp(Ω), (24) Z

Ω

(φ(x) + λ1)|y(x)|^pdx = 0. (25)

Now we show that y 6= 0. Indeed, from the results obtained it follows that kDynk^p_Lp(Ω;R^N)− λ1kynk^p_Lp(Ω)→ 0

and by the compactness of the imbedding W₀^1,p(Ω) ⊂ L^p(Ω) we get kynkL^p(Ω)→ kykL^p(Ω).

Since kDynk_L^p_(Ω;R^N)≥ ckynk_W^1,p

0 (Ω)= c, c > 0 (the equivalence of the norms), we arrive at λ1kyk^p_Lp(Ω)≥ c^p which establishes the assertion. Therefore, taking into account (24) we conclude that y 6= 0 is an λ1-eigenfunction. Since φ(x) >

−λ1 on a set of positive measure (by (H5)) and |y(x)| > 0 for a.e. x ∈ Ω

(see Lindqvist, 1990), we are led to the contradiction with (25). The proof of Lemma 3.2 is complete.

If we strengthen the hypotheses (H4) and (H5) as shown below then the statements of Lemma 3.1 and Lemma 3.2 still hold true. This is the case when the Ambrosetti-Rabinowitz type conditions (H2)(i)and (H2)(ii)are redundant.

Lemma 3.3 Assume the hypotheses (H1), (H3). Moreover, assume the follow- ing:

(H4)1 The classical growth condition: There exists a constant κ > 0 such that

|∂ξj(x, ξ)| ≤ κ(1 + |ξ|^p−1), ∀ ξ ∈ R and for a.e. x ∈ Ω;

(H5)1The inequality holds:

pj(x, ξ) ≥ φ(x)|ξ|^p, ∀ ξ ∈ R and a.e. x ∈ Ω, with φ(x) ∈ L^∞(Ω) and φ(x) > −λ1 a.e. in Ω.

Moreover, suppose that for a sequence {un} ⊂ W₀^1,p(Ω) ∩ L^∞(Ω) there exists εnց 0 such that the conditions below are fulfilled:

Z

Ω

|Dun(x)|^p−2

Dun(x), Dv(x) − Dun(x)

R^Ndx +

Z

Ω

j⁰ x, un(x); v(x) − un(x)

dx ≥ −εnkv − unk_W^1,p

0 (Ω),

∀ v ∈ Lin({un, θ}), (26) and

  1 p

Z

Ω

|Dun(x)|^pdx + Z

Ω

j x, un(x)
dx

 ≤ C, C > 0. (27)

where Lin({un, θ}) is the linear subspace of W₀^1,p(Ω) spanned by {θ, un}. Then the sequence {un} is bounded in W₀^1,p(Ω), i.e. there exists M > 0 such that

kunk_W^1,p

0 (Ω)≤ M. (28)

Moreover, there exists ρ > 0 such that R(u) :=¹_pkDuk^p_Lp(Ω;R^N)+

Z

Ω

j(u) dx ≥ η, η = const > 0, (29)

is valid for any u ∈ W₀^1,p(Ω) with kuk_W^1,p

0 (Ω)= ρ.

Proof. Let us begin with (28). Suppose on the contrary that the claim is not true, i.e. there exists a sequence {un}^∞_n=1 ⊂ W₀^1,p(Ω) ∩ L^∞(Ω) with

kunk_W^1,p

0 (Ω) → ∞ for which (26) and (27) hold. Combining (27) multiplied by any µ > p with (26) (with v = 2un substituted) yields

µC + εnkunk_W^1,p

0 (Ω)≥^µ−p_p 

kDunk^p_Lp(Ω;R^N)− λ1kunk^p_Lp(Ω)



+ Z

Ω

µj(un) − j⁰(un; un) + λ1µ−p p |un|^p

dx. (30)

In view of (H4)1, −j⁰(un; un) ≥ −k|un|^p− k for some k > 0, so by (5) we obtain µC + εnkunk_W^1,p

0 (Ω)≥ c^µ−p_p kD(bun)k^p_Lp(Ω;R^N)

+ Z

Ω

µj(un) − j⁰(un; un) + λ1µ−p p |un|^p

dx

≥ c^µ−p_p kD(bun)k^p_Lp(Ω;R^N)+ Z

Ω

µ

p(φ + λ1) − λ1− k

|un|^pdx − k|Ω|.

≥ c^µ−p_2p kD(bun)k^p_Lp(Ω;R^N)+ c^µ−p_2p 

kD(bun)k^p_Lp(Ω;R^N)− λ1kbunk^p_Lp(Ω)



+ Z

Ω

_µ

p(φ + λ1) − λ1− k

|un|^p+ c^µ−p_2p λ1|bun|^p

dx − k|Ω|,

|Ω| being the Lebesgue measure of Ω. Now we state the estimate that will be useful for our further investigations:

µC + k|Ω| + εnkunk_W^1,p

0 (Ω)≥ c^µ−p_2p kD(bun)k^p_Lp(Ω;R^N)

+ Z

Ω

_µ

p(φ + λ1) − λ1− k

|un|^p+ c^µ−p_2p λ1|bun|^p dx

≥ c^µ−p_2p kD(bun)k^p_Lp(Ω;R^N)

− Z

Ω⁻µ

λ1+ k −^µ_p(φ + λ1)

|un|^pdx + c^µ−p_2p λ1

Z

Ω⁻µ

|bun|^p

dx, (31)

where Ω⁻_µ := {x ∈ Ω : ^µ_p(φ + λ1) < λ1+ k}. Notice that due to φ + λ1> 0 a.e.

in Ω it follows that |Ω⁻_µ| → 0 as µ → +∞. This (by |a ± b|^p≤ 2^p−1(|a|^p+ |b|^p), a, b ∈ R) implies

µC + k|Ω| + εnkunk_W^1,p

0 (Ω)≥ c^µ−p_2p kD(bun)k^p_Lp(Ω;R^N)

+ Z

Ω⁻µ

c^µ−p_2p λ1+ 2^{p−1 µ}_p(φ + λ1) − λ1− k


|bun|^pdx

−e^p_n2^p−1 Z

Ω⁻µ

λ1+ k − ^µ_p(φ + λ1)

|θ|^pdx.

Since φ + λ1> 0, one can choose µ large enough to get c^µ−p_2p λ1+ 2^{p−1 µ}_p(φ + λ1) − λ1− k

≥ 0. (32)

This yields the estimate µC + k|Ω| + εn(kbunk_W^1,p

0 (Ω)+ en) ≥ c^µ−p_2p kD(bun)k^p_Lp(Ω;R^N)

−e^p_n Z

Ω⁻µ

λ1+ k −^µ_p(φ + λ1)

|θ|^pdx,

which allows the conclusion that en → ∞. Otherwise, we would have the boundedness of {bun} and consequently the boundedness of {un} in W₀^1,p(Ω), contrary to our supposition. Dividing (31) by e^p_n yields

µC+k|Ω|

e^pn + ^εn

e^p−1n k^ub_eⁿ

nk_W^1,p

0 (Ω)+ 1

≥ c^µ−p_2p kD(^ub_eⁿ

n)k^p_Lp(Ω;R^N)

− Z

Ω⁻_µ

λ1+ k −^µ_p(φ + λ1)

|^bu_eⁿ

n + θn|^p

dx + c^µ−p_2p λ1

Z

Ω⁻_µ

|^ub_eⁿ

n|^pdx.

Hence

µC+k|Ω|

e^pn + ^εn

e^p−1n k^ub_eⁿ

nk_W^1,p

0 (Ω)+ 1

≥ c^µ−p_2p kD(^ub_eⁿ

n)k^p_Lp(Ω;R^N)

+ Z

Ω⁻µ

2^p−1(−λ1− k +^µ_p(φ + λ1)

+ c^µ−p_2p λ1

|^ub_enⁿ|^pdx

−2^p−1 Z

Ω⁻µ

λ1+ k −^µ_p(φ + λ1)

|θ|^p dx.

Choosing µ like in (32) gives rise to

µC+k|Ω|

e^pn + ^εn

e^p−1n k^ub_eⁿ

nk_W^1,p

0 (Ω)+ 1

≥ c^µ−p_2p kD(^ub_eⁿ

n)k^p_Lp(Ω;R^N)

−2^p−1 Z

Ω⁻µ

λ1+ k − ^µ_p(φ + λ1)

|θ|^p dx,

which means that {^ub_enⁿ} is bounded in W₀^1,p(Ω). We claim that, in fact, ^bu_enⁿ → 0 strongly in W₀^1,p(Ω). Indeed, because |Ω⁻_µ| → 0 as µ → +∞, for an arbitrary ǫ > 0 one can choose µ sufficiently large, say µ ≥ µ0, so that

2^p−1 Z

Ω⁻µ

λ1+ k −^µ_p(φ + λ1)

|θ|^pdx ≤ ^ǫ₂. For such a µ one can find n0large enough to fulfill

µC+k|Ω|

e^pn +_e^εp−1ⁿ

n k^ub_eⁿ_nk_W^1,p

0 (Ω)+ 1

≤₂^ǫ, n ≥ n0. Therefore we are led to the estimate

ǫ ≥ c^µ−p_2p kD(^bu_eⁿ

n)k^p_Lp(Ω;R^N)≥ ckD(^ub_eⁿ

n)k^p_Lp(Ω;R^N), µ ≥ max{3p, µ0}, n ≥ n0, which establishes the strong convergence^ub_enⁿ → 0 in W₀^1,p(Ω). Proceeding like in the proof of Lemma 3.1 we get (28). For (29) it is sufficient to invoke Lemma 3.2.

From now on we shall assume the hypothesis:

(H6) R

Ωj(x, 0) dx ≤ 0 and there exists e ∈ W₀^1,p(Ω) ∩ L^∞(Ω), e 6= 0, such that R(se) ≤ 0, ∀ s ≥ 1.

Lemma 3.4 Assume that (H1)-(H2) are satisfied, R

Ωj(x, 0) dx ≤ 0 and for some θ ∈ V0, θ 6= 0,

lim inf

s→+∞

Z

Ω

j x, sθ(x)

+^λ_p¹s^p|θ(x)|^pdx < 0. (33) Then (H6) holds.

Proof. The assertion easily holds for e = s0θ with sufficiently large s0> 0.

Lemma 3.5 Assume that (H1) is fulfilled and instead of (H2) the stronger hy- pothesis is satisfied:

(H2)^′ One of the two conditions below holds (the Ambrosetti-Rabinowitz condi- tions):

(i) There exist µ > p, 1 ≤ σ < p, a ∈ L¹(Ω) and a constant k ≥ 0 such that

µj(x, ξ) − j⁰(x, ξ; ξ) ≥ −a(x) − k|ξ|^σ, ∀ ξ ∈ R and for a.e. x ∈ Ω;

(ii) There exist 0 < ν < p, 1 ≤ σ < p, a ∈ L¹(Ω) and a constant k ≥ 0 such that

−νj(x, ξ)−j⁰(x, ξ; −ξ) ≥ −a(x)−k|ξ|^σ, ∀ ξ ∈ R and for a.e. x ∈ Ω, Moreover, assume that R

Ωj(x, 0) dx ≤ 0 and for some v0 ∈ W₀^1,p(Ω) ∩ L^∞(Ω) (Motreanu and Panagiotopoulos, 1999),

lim inf

s→+∞s^−σ Z

Ω

j x, sv0(x)

dx < k

σ − µkv0k^σ_Lσ(Ω), (34) with the positive constants k, µ, σ entering (H2). Then (H6) holds.

Proof. We follow the lines of Motreanu and Panagiotopoulos (1999). For all τ 6= 0, x ∈ Ω and ξ ∈ R, the formula below of generalized gradient (with respect to τ ) holds

∂τ(τ^−µj(x, τ ξ)) = τ^−µ−1[−µj(x, τ ξ) + ∂ξj(x, τ ξ)(τ ξ)],

for the constant µ > p fulfilling (H2). Since the function τ 7→ τ^−µj(x, τ ξ) is differentiable a.e. on R, the equality above and a classical property of Clarke’s generalized directional derivative imply that

t^−µj(x, tξ) − j(x, ξ) = Z t

1

d

dτ(τ^−µj(x, τ ξ))dτ

≤ Z t

1

τ^−µ−1[−µj(x, τ ξ) + j⁰(x, τ ξ; τ ξ)]dτ, ∀ t > 1, a.e. x ∈ Ω, ξ ∈ R.