Page 1

arXiv:1103.0163v1 [math.AP] 1 Mar 2011

MOUNTAIN PASS SOLUTIONS FOR QUASI-LINEAR

EQUATIONS VIA A MONOTONICITY TRICK

BENEDETTA PELLACCI AND MARCO SQUASSINA

Abstract. We obtain the existence of symmetric mountain pass solutions for quasi-

linear equations without the typical assumptions which guarantee the boundedness of an

arbitrary Palais-Smale sequence. This is done through a recent version of the monotonicity

trick proved in [20]. The main results are new also for the p-Laplacian operator.

1. Introduction

Let N > p > 1. In the study of the quasi-linear partial differential equation

(1.1)− div(jξ(u,Du)) + js(u,Du) + V (x)|u|p−2u = g(u),u ∈ W1,p(RN)

by means of variational methods, a rather typical assumption on j(s,ξ) and g(s) is that

there exist p < q < Np/(N − p), δ > 0 and R ≥ 0 such that

(1.2)qj(s,ξ) − js(s,ξ)s − (1 + δ)jξ(s,ξ) · ξ − qG(s) + g(s)s ≥ 0,

for all s ∈ R such that |s| ≥ R and any ξ ∈ RN(cf. [2,6]). This condition ensures that every

Palais-Smale sequence, in a suitable sense, of the associated functional f : W1,p(RN) → R,

f(u) =

?

RNj(u,Du) +1

p

?

RNV (x)|u|p−

?

RNG(u),

is bounded in W1,p(RN). We might refer to this technical condition as the generalized

Ambrosetti-Rabinowitz condition, involving the terms of the quasi-linear operator j. In

fact, in the treatment of the non-autonomous semi-linear equation

(1.3)− ∆u + V (x)u = g(u),u ∈ H1(RN),

the previous inequality (1.2) reduces to the classical Ambrosetti-Rabinowitz condition [1],

namely 0 < qG(s) ≤ g(s)s, for every s ∈ R with |s| ≥ R. Of course, aiming to achieve

the existence of multiple solutions for equation (1.1), one needs to know that the Palais-

Smale condition for f is satisfied at an arbitrary energy level, and hence it is necessary

to guarantee that Palais-Smale sequences are always at least bounded, through condition

(1.2). On the contrary, under suitable assumptions, if one merely focuses on the existence

of a nonnegative Mountain Pass solution of (1.1), it is reasonable to expect that by a clever

selection of a special Palais-Smale sequence at the Mountain Pass level c one could reach

2000 Mathematics Subject Classification. 74G65; 35J62; 35A15; 35B06; 58E05.

Key words and phrases. Non-smooth critical point theory, monotonicity trick, Palais-Smale condition.

The second author was partially supported by 2007 MIUR Project: Metodi Variazionali e Topologici

nello Studio di Fenomeni non Lineari.

1

Page 2

2 BENEDETTA PELLACCI AND MARCO SQUASSINA

the goal of getting a solution to (1.1) without knowing that the Palais-Smale condition

holds. The existence of such a nice sequence is possible since the definition of c allows

to detect continuous paths γ : [0,1] → W1,p(RN) with a very good behavior. The idea,

considering for instance problems (1.3), is to see f = f1as the end point of the continuous

family of C1functionals fλ: H1(RN) → R,

fλ(u) =1

2

?

RN|Du|2+1

2

?

RNV (x)|u|2− λ

?

RNG(u).

When fλ satisfies a uniform Mountain Pass geometry, then it is possible to use the so

called monotonicity trick for C1smooth functionals, originally discovered by Struwe [22]

in a very special setting and generalized and formalized later in an abstract framework by

Jeanjean [11] and Jeanjean-Toland [13]. This strategy provides a bounded Palais-Smale

sequence for all λ fixed, up to a set of null measure. Then, by requiring some compactness

condition one can detect a sequence (λj), increasingly converging to 1, for which there

corresponds a sequence (uλj) of solutions to (1.3) at the Mountain Pass level cλj, namely

(1.4)cλ= inf

γ∈Γsup

t∈[0,1]fλ(γ(t)), Γ = {γ ∈ C([0,1],W1,p(RN)) : γ(0) = 0,γ(1) = w},

being w ∈ W1,p(RN) a suitable function with fλ(w) < 0 for any value of λ. Then, being

uλjexact solutions, one can exploit the Pohˇ ozaev identity and combine it with the energy

level constraint to show in turn that (uλj) is a bounded Palais-Smale sequence for f1. In

the case of semi-linear equations such as (1.3), we refer the reader to [3,12] where the

approach has been successfully developed.

The main goal of this manuscript is twofold. On one hand, we intend to show how

condition (1.2) can be completely removed by using a general version of the monotonicity

trick recently developed in [20] in the framework of the non-smooth critical point theory

of [7,8]. In this respect, first, in order to analyze the most clarifying concrete situation, we

consider a class of functionals invariant under orthogonal transformations, set in the space

of radial functions (see Theorem 1.1). As in the smooth case, by studying a penalized

functional fλwe will obtain a sequence of λjconverging to one, with corresponding weak

solutions uλj. In order to obtain that the sequence (uλj) is bounded, a general version

of the Pohˇ ozaev identity [9] for merely C1weak solutions will be crucial, as C1,αis the

optimal regularity if p ?= 2 [23]. Moreover, a generalized version of the Palais’ symmetric

criticality principle recently achieved in [19] will be exploited. These results are new also

in the particular meaningful case j(u,Du) = |Du|p/p with p ?= 2, being the case p = 2

covered in [3]. On the other hand, when one does not restrict the functional to the space of

radially symmetric functions (see Theorem 1.2), it is possible to make a stronger use of the

result in [20] to construct a bounded, almost symmetric (cf. (3.1)), Palais-Smale sequence

which will give a radial and radially decreasing solution. At the high level of generality of

equation (1.1), proving a priori that the radial solution is decreasing seems a particularly

strong fact. These results are new also for j(u,Du) = |Du|p/p, even with p = 2.

Let us now state the main results of the paper. Let N > p > 1 and let j : R×R+→ R+

be a C1function such that the map t ?→ j(s,t) is increasing and strictly convex. Moreover,

Page 3

MOUNTAIN PASS SOLUTIONS FOR QUASI-LINEAR EQUATIONS3

we assume that there exist α,β > 0 with

αtp≤ j(s,t) ≤ βtp,

|js(s,t)| ≤ βtp,

for every s ∈ R and t ∈ R+,

|jt(s,t)| ≤ βtp−1,

for every s ∈ R and t ∈ R+.

(1.5)

for every s ∈ R and t ∈ R+, (1.6)

js(s,t)s ≥ 0,(1.7)

Let V : R+→ R+be a C1function such that there exist m,M ∈ R+with

(1.8)0 < m ≤ V (τ) ≤ M,for every τ ∈ R+.

Furthermore, we shall assume that

(1.9)?V′(|x|)|x|?LN/p(RN)< αpS,

where α is the number appearing in (1.5) and S is the best Sobolev constant. Apart from

the natural growths (1.5)-(1.6), conditions (1.7) is a typical requirement in the frame of

quasi-linear equations, which helps [2,6,16,18,21] in the achievement of both existence and

summability issues related to equation (1.1). Under (1.5) and (1.8), the functional defined

either in W1,p

rad(RN) or in W1,p(RN) as

u ?→

?

RNj(u,|Du|)+ V (|x|)|u|p

p

,

is continuous but not even locally Lipschitz, as it can be easily checked. Moreover, it admits

Gateaux derivatives along any bounded direction v, but not on an arbitrary direction v

of either W1,p

machinery developed in [7, 8] for continuous functionals, the related monotonicity trick

proved in [20] and the Palais’ symmetric criticality principle formulated in [19].

rad(RN) or W1,p(RN). This is the reason why we will make use of the abstract

Let p∗:= Np/(N − p) and consider the equation

jt(u,|Du|)Du

(1.10)− div

?

|Du|

?

+ js(u,|Du|) + V (|x|)up−1= g(u) in RN.

Our first main result is the following

Theorem 1.1. Assume (1.5)-(1.9) and let g : R+→ R+be continuous with g(0) = 0 and

extended by zero on R−. Moreover,

(1.11) lim

s→0+

g(s)

sp−1= lim

s→+∞

g(s)

sp∗−1= 0,

and, furthermore, for G(s) =?s

(1.12)

0g(t),

there exists s > 0 such that pG(s) − Msp> 0.

Then equation (1.10) admits a nontrivial, nonnegative, distributional and radially symmet-

ric solution u ∈ W1,p(RN).

This result seems new even in the particular p-Laplacian case j(s,t) = tp/p with p ?= 2. In

order to prove Theorem 1.1, we consider the continuous functionals fλ: W1,p

rad(RN) → R,

(1.13)fλ(u) =

?

RNj(u,|Du|) +

?

RNV (|x|)|u|p

p

− λ

?

RNG(u),λ ∈ [δ,1],

Page 4

4 BENEDETTA PELLACCI AND MARCO SQUASSINA

for some suitable value of δ ∈ (0,1). First we shall prove that fλfulfills a uniform Mountain

Pass geometry. Next we show that for all λ ∈ (δ,1] any bounded Palais-Smale sequence

is, actually, strongly convergent. Furthermore, by applying the monotonicity trick of [20]

and the Palais’ symmetric criticality principle proved in [19] for continuous functionals,

a sequence λh⊂ [δ,1) with λhր 1 is detected such that, for each h ≥ 1, there exists a

distributional solution uλh∈ W1,p

rad(RN) of

−div

?

jt(u,|Du|)Du

|Du|

?

+ js(u,|Du|) + V (|x|)up−1= λhg(u) in RN

at the Mountain Pass level cλh. Then, by exploiting a Pohˇ ozaev identity [9] for C1solutions

of (1.10), we show in turn that (uλh) is also a bounded Palais-Smale condition for f1, and

passing to the limit will provide the desired conclusion.

Our second main result is the following

Theorem 1.2. Assume (1.5)-(1.9), let g : R+→ R+be continuous with g(0) = 0, extended

by zero on R−, satisfying (1.12), and such that for all ε > 0 there is Cε∈ R+with

(1.14)|g(s)| ≤ εsp−1+ Cεsq−1, p < q < p∗,

for every s ∈ R+. Let V also satisfy

(1.15)|x| ≤ |y| =⇒ V (|x|) ≤ V (|y|)

for every x,y ∈ RN.

Then equation (1.10) admits a nontrivial, nonnegative, distributional, radially symmetric

and decreasing solution u ∈ W1,p(RN).

This result seems new even in the particular p-Laplacian case j(s,t) = tp/p, included

p = 2. In place of (1.11), here we need the slightly more restrictive condition (1.14),

since we cannot work directly on sequences of radial functions, which enjoy uniform decay

properties. In order to prove Theorem 1.2, we argue on the continuous functionals fλ:

W1,p(RN) → R again defined as in (1.13) for all λ ∈ (δ,1], for a suitable δ ∈ (0,1).

Hence here we do not restrict the functional to the space of radially symmetric functions.

However, we still proceed as indicated above for the proof Theorem 1.1, but, by exploiting

the symmetry properties of the functional under polarization (cf. [20]) we use the symmetry

features of the monotonicity trick of [20] and we obtain the existence of a bounded and

almost symmetric (cf. (3.1)) Palais-Smale sequence for f1. Possessing a compactness result

for such sequences, we can conclude the proof. We remark that in this second statement the

solution found is not only radially symmetric, but also automatically radially decreasing.

While in Theorem 1.1 the solution is found at the restricted Mountain Pass level

crad= inf

γ∈Γradsup

t∈[0,1]f1(γ(t)),Γrad= {γ ∈ C([0,1],W1,p

rad(RN)) : γ(0) = 0,γ(1) = w},

in Theorem 1.2 the solution is found at the global Mountain Pass level

c = inf

γ∈Γsup

t∈[0,1]f1(γ(t)), Γ = {γ ∈ C([0,1],W1,p(RN)) : γ(0) = 0,γ(1) = w}.

Page 5

MOUNTAIN PASS SOLUTIONS FOR QUASI-LINEAR EQUATIONS5

Of course, on one hand, we have c ≤ crad. On the other hand it is not clear if, in general,

one has c = crador c < cradalthough, precisely as a further consequence of Theorem 1.2,

this occurs when V is constant and the map t ?→ j(s,t) is p-homogeneous (see Remark 3.4).

2. Proof of Theorem 1.1

We will prove Theorem 1.1 by studying the functionals fλ : W1,p

(1.13). Taking into account assumptions (1.5), (1.8) and (1.11), recalling [4, Theorem

A.VI], it follows that fλis well defined and (merely) continuous. In turn, we shall exploit

the non-smooth critical point theory of [7, 8] including the connection between critical

points in a suitable sense and solutions of the associated Euler’s equation (see for instance

[16, Theorem 3] and also [19, Proposition 6.16] for the symmetric setting). More precisely

under assumption (1.5)-(1.9), the critical points of fλare distributional solutions of

rad(RN) → R defined in

(2.1)− div

?

jt(u,|Du|)Du

|Du|

?

+ js(u,|Du|) + V (|x|)|u|p−2u = λg(u) in RN.

Combining the following two lemmas shows that the minimax class (1.4) is nonempty and

that the family (fλ) enjoys a uniform Mountain Pass geometry whenever λ varies inside

the interval [δ0,1], for a suitable δ0> 0.

Lemma 2.1. Assume (1.5), (1.8) and (1.11)-(1.12). Then there exists δ0 ∈ (0,1) and

a curve γ ∈ C([0,1],W1,p

λ ∈ [δ0,1].

rad(RN)), independent of λ, such that fλ(γ(1)) < 0, for every

Proof. Due to (1.12), there exists z ∈ W1,p

rad(RN), z ≥ 0 and Schwartz symmetric, such that

G(z) −M

pzp?

?

RN

?

> 0.

To see this, follow closely the first part of [4, Step 1, pp.324-325]. In turn, let δ0∈ (0,1)

with

?

RN

(2.2)

?

δ0G(z) −M

pzp?

> 0,

and define the curve η ∈ C([0,∞),W1,p

η(0) := 0. From (1.5) and (1.8) it follows that

rad(RN)) by setting η(t) := z(·/t) for t ∈ (0,∞) and

fλ(η(t)) ≤ βtN−p?Dz?p

Lp(RN)− tN

?

RN

?

δ0G(z) −M

pzp?

,

yielding, on account of (2.2), a time t0> 0 such that fλ(η(t0)) < 0 for every λ ∈ [δ0,1].

Then, the curve γ ∈ C([0,1],W1,p

the required property and Γ is nonempty by taking w := γ(1).

rad(RN)), independent of λ, defined by γ(t) := η(t0t) has

?

Lemma 2.2. Assume (1.5), (1.8) and (1.11). Let δ0> 0 be the number found in Lemma 2.1.

There exist σ > 0 and ρ > 0, independent of λ, such that fλ(u) ≥ σ for any u in W1,p

with ?u?1,p= ρ and for every λ ∈ [δ0,1].

rad(RN)

Page 6

6 BENEDETTA PELLACCI AND MARCO SQUASSINA

Proof. Condition (1.11) implies that for every ε > 0, there exists Cεsuch that

(2.3)|g(s)| ≤ εsp−1+ Cεsp∗−1, for every s ∈ R+.

Then, fixed ε0< m, we find Cε0such that for every λ ∈ [δ0,1]

fλ(u) ≥ α?Du?p

Lp(RN)+m − ε0

p

?u?p

Lp(RN)− Cε0?u?p∗

W1,p(RN).

This last inequality immediately gives the conclusion.

?

We will use the following compactness condition.

Definition 2.3. Let λ,c ∈ R. We say that fλsatisfies the concrete-(BPS)ccondition if

any bounded sequence (uh) ⊂ W1,p

rad(RN) such that there is wh∈ W−1,p′

rad (RN) with

(2.4)fλ(uh) → c,?f′

λ(uh),v? = ?wh,v?

for every v ∈ C∞

c,rad(RN),

and wh→ 0

admits a strongly convergent subsequence.

In the next result we will use the property

(2.5)jt(s,t)t ≥ αtp,

which can be obtained by hypotheses (1.5) once one has observed that, as j is a strict

convex function with respect to t, it results 0 = j(s,0) ≥ j(s,t) + jt(s,t) · (0 − t).

Proposition 2.4. Let λ ∈ [δ0,1], c ∈ R and assume (1.5)-(1.8) and (1.11). Then the

functional fλsatisfies the concrete-(BPS)c.

Proof. Let (uh) ⊂ W1,p

Then, in turn, there exists a subsequence, still denoted by (uh), converging weakly in

W1,p

u ∈ W1,p

Du almost everywhere. More precisely, since the variational formulation is here restricted

to radial functions, this property follows by arguing as in [19, proof of Theorem 6.4]. Then,

it is possible to follow the same arguments used in [16, Step 2 of Lemma 2] (see also [18])

for bounded domains, in order to pass to the limit in the equation in (2.4) and obtain in

turn that u satisfies the variational identity

rad(RN) be a bounded sequence which satisfies the properties in (2.4).

rad(RN), strongly in Lq(RN) for any q ∈ (p,p∗) and almost everywhere to a function

rad(RN). Moreover, we can apply the result in [5] to obtain that Duhconverges to

?

RNjt(u,|Du|)Du

?

|Du|· Dϕ +

?

RNjs(u,|Du|)ϕ

?

+

RNV (|x|)|u|p−2uv = λ

RNg(u)ϕ,∀ϕ ∈ C∞

c,rad(RN).

In fact, all the particular test functions built in [16,18] to achieve this identity are radial,

since each uh is radial and ϕ is a fixed radial function. Observe also that a function

ϕ ∈ W1,p

C∞

rad(RN)∩L∞(RN) can be approximated, in the ?·?1,pnorm, by a sequence (ϕm) ⊂

c,rad(RN) with ?ϕm?L∞ ≤ c(ϕ), for some positive constant c(ϕ). Whence, exploiting

Page 7

MOUNTAIN PASS SOLUTIONS FOR QUASI-LINEAR EQUATIONS7

(1.6)-(1.8) and (1.11), recalling that u is radial and arguing as in [16, Proposition 1], it

follows that u is an admissible test function, namely

?

(2.6)

RNjt(u,|Du|)|Du|+

?

RNjs(u,|Du|)u+

?

RNV (|x|)|u|p= λ

?

RNg(u)u.

Furthermore, taking into account that uh∈ W1,p

can use [4, Theorem A.I] to obtain that

rad(RN) and exploiting conditions (1.11), we

lim

h→∞

?

RNg(uh)uh=

?

RNg(u)u.

Observe that, applying by Fatou’s lemma in view of (1.7)-(1.8) and (2.5), formula (2.6)

implies

?

h→∞

RNjt(u,|Du|)|Du|+ V (|x|)|u|p≤ liminf

??

??

RNjt(uh,|Duh|)|Duh| + V (|x|)|uh|p?

≤ limsup

h→∞

RNjt(uh,|Duh|)|Duh| + V (|x|)|uh|p?

≤ −liminf

h→∞

?

RNjs(uh,|Duh|)uh+ lim

?

h→∞λ

?

RNg(uh)uh

= −

?

RNjs(u,|Du|)u+ λ

RNg(u)u

=

?

RNjt(u,|Du|)|Du|+ V (|x|)|u|p.

Then, taking into account (1.8) and (2.5), it results

lim

h→∞

?

RN|Duh|p+ m|uh|p=

?

RN|Du|p+ m|u|p,

giving the desired convergence of (uh) to u via the uniform convexity of W1,p(RN).

?

Next, we state the main technical tool for the proof of the first theorem.

Lemma 2.5. Assume that conditions (1.5)-(1.8) and (1.11)-(1.12) hold and that fλsat-

isfies the concrete-(BPS)cfor all c ∈ R and all λ ∈ [δ0,1]. Then there exists a sequence

(λj,uj) ⊂ [δ0,1] × W1,p

jt(u,|Du|)Du

|Du|

rad(RN) with λjր 1 and where ujis a distributional solution to

(2.7)− div

?

?

+ js(u,|Du|)+ V (|x|)|u|p−2u = λjg(u)

in RN,

such that fλj(uj) = cλj.

Proof. The result follows by applying [20, Corollary 3.3] to the minimax class defined in

(1.4), with the choice of spaces X = S = V = W1,p

u∗:= u as the identity maps. In fact, assumptions (H1) and (H2) are fulfilled thanks to

Lemmas 2.1 and 2.2. Condition (H3) is implied by the structure of fλas it can be verified by

a straightforward direct computation. Finally assumption (H4) is evidently satisfied since

uHis the identity map. Since X = W1,p

(uj) provided by [20, Corollary 3.3] are distributional with respect to test functions in

rad(RN) and by defining uH:= u and

rad(RN), it turns out that, a priori, the solutions

Page 8

8 BENEDETTA PELLACCI AND MARCO SQUASSINA

C∞

function in C∞

c,rad(RN). The fact that ujis, actually, a distributional solution with respect to any test

c(RN) follows by [19, Theorem 4.1 and end of the proof of Theorem 6.4].

?

Proposition 2.6. Assume (1.5), (1.8) and (1.11)-(1.12).

increasing and continuous from the left.

The map λ → cλ is non-

Proof. The fact that cλis non-increasing trivially follows from the fact that G ≥ 0. The

proof of the left-continuity follows arguing by contradiction exactly as done in [11, Lemma

2.3].

?

2.1. Proof of Theorem 1.1 concluded. Proposition 2.4 allows us to apply Lemma 2.5

and obtain, in turn, a sequence uj of distributional solution of (2.7) at the energy level

cλj. Following the argument in [10, Lemma 4.1] and applying [17, Theorem 1 and Remark

p.261] one obtains uj∈ L∞

uj∈ C1,α(RN). As a consequence, we can apply the Pohˇ ozaev variational identity for C1

solutions of equation (2.7) stated in [9, Lemma 1], by choosing therein h(x) = hk(x) =

H(x/k)x ∈ C1

H(x) = 0 for |x| ≥ 2. Letting k → ∞ and taking into account conditions (1.5), (1.6) and

that V′(|x|)|x| ∈ LN/p(RN), we reach

loc(RN) and then, via standard regularity arguments (see [15])

c(RN;RN), where H ∈ C1

c(RN) is such that H(x) = 1 on |x| ≤ 1 and

?

RNjt(uj,|Duj|)|Duj| − N

?

?

RNj(uj,|Duj|) −N

p

?

RNV (|x|)|uj|p

+ Nλj

RNG(uj) −1

p

?

RNV′(|x|)|x||uj|p= 0, for all j ≥ 1.

In turn, each ujsatisfies the following identity

fλ(uj) =1

N

?

RNjt(uj,|Duj|)|Duj| −

1

Np

?

RNV′(|x|)|x||uj|p, for all j ≥ 1.

Since fλ(uj) = cλjand recalling (2.5) one has

?Duj?p

Lp(RN)

?αpS − ?V′(|x|)|x|?LN/p(RN)

?≤ pNScλj,for all j ≥ 1,

where S is the best constant for the Sobolev embedding. The last inequality, jointly with

(1.9) and Proposition 2.6, yields the existence of A > 0 such that

(2.8)?Duj?Lp(RN)≤ A,for all j ≥ 1.

Also, since ujsolves (2.7), by testing it with ujitself (which is admissible), (1.7) and (2.5)

give

?

So that, conditions (1.8), (2.3) and (2.8) yield, for any fixed ε < m,

RNV (|x|)|uj|p− λj

?

RNg(uj)uj≤ 0.

(2.9)(m − λjε)?uj?p

Lp(RN)≤ λj

Cε

Sp∗/pAp∗.

Since (λj) is bounded, by combining (2.8) and (2.9) we get that (uj) is bounded in

W1,p

In turn, let us observe that (uj) is a concrete-(BPS)c1for the functional

rad(RN).

Page 9

MOUNTAIN PASS SOLUTIONS FOR QUASI-LINEAR EQUATIONS9

f1. In fact notice that, taking into account that G(uj) remains bounded in L1(RN) due to

inequality (2.3), that fλj(uj) = cλjand recalling Proposition 2.6, it follows as j → ∞

?

(2.10)f1(uj) = fλj(uj) + (λj− 1)

RNG(uj) = cλj+ (λj− 1)

?

RNG(uj) = c1+ o(1).

Furthermore, by defining ˆ wj= (λj−1)g(uj) ∈ W−1,p′(RN), for every v ∈ C∞

?

?

c(RN) we have

?f′

1(uj),v? =

RNjt(uj,|Duj|)Duj

|Duj|· Dv +

?

RNjs(uj,|Duj|)v (2.11)

+

RNV (|x|)|uj|p−2ujv −

?

RNg(uj)v = ?f′

λj(uj),v? + ?ˆ wj,v? = ? ˆ wj,v?.

Then, since in light of (2.3) and (2.8)-(2.9), ˆ wj→ 0 in W−1,p′(RN) as j → ∞, Proposition

2.4 applied to f1 and with c = c1 implies that there exists a function u ∈ W1,p

such that, up to a subsequence, (uj) converges to u strongly in W1,p

of formulas (2.10)-(2.11) and the continuity of f1, and by an application of Lebesgue’s

Theorem we conclude that u is a nontrivial radial Mountain Pass solution of (1.10). Finally,

u is automatically nonnegative, as follows by testing (1.10) with the admissible (by [18,

Proposition 3.1] holding also for unbounded domains) test function −u−, in view of (1.7),

(2.5) and the fact that g(s) = 0 for every s ≤ 0.

rad(RN)

rad(RN). On account

3. Proof of Theorem 1.2

Equation (1.10) is investigated by studying the continuous functional fλ: W1,p(RN) → R

with fλ(u) again defined as in (1.13) which, for λ = 1, corresponds to the action functional

associated to (1.10).

Definition 3.1. Let λ ∈ [δ0,1], for some δ0> 0, and c ∈ R. We say that fλsatisfies the

concrete-(SBPS)ccondition if every bounded sequence (uh) in W1,p(RN) such that there

exists wh∈ W−1,p′(RN) with wh→ 0 as h → ∞,

fλ(uh) → c,?f′

λ(uh),v? = ?wh,v?∀v ∈ C∞

c(RN),

and

(3.1)?uh− u∗

h?Lp(RN)∩Lp∗(RN)→ 0,

admits a strongly convergent subsequence. Here u∗:= |u|∗, where ∗ denoted the Schwarz

symmetrization.

Proposition 3.2. Let λ ∈ [δ0,1], for some δ0> 0, c ∈ R and assume that (1.5)-(1.8) and

(1.14) hold. Then the functional fλsatisfies the concrete-(SBPS)c.

Proof. Given a concrete-(SBPS)csequence (uh) ⊂ W1,p(RN), as in the proof of Propo-

sition 2.4, up to a subsequence, (uh) converges to a u weakly, almost everywhere and, in

addition, Duhconverges to Du almost everywhere. The main difference with respect to

Proposition 2.4 is that the crucial limit

?

(3.2)lim

h

RNg(uh)uh=

?

RNg(u)u,

Page 10

10 BENEDETTA PELLACCI AND MARCO SQUASSINA

admits now a different justification.

W1,p(RN), then (u∗

Therefore, since for every p < q < p∗the injection map i : W1,p

pletely continuous, up to a subsequence, it follows that u∗

some z ∈ Lq(RN), for p < q < p∗. Due to ?uh− u∗

Lq(RN), as

?uh− z?Lq(RN)≤ C?uh− u∗

Of course z = u, allowing to conclude that

Since (u∗

h) ⊂ W1,p

rad(RN) and (uh) is bounded in

h) is bounded in W1,p(RN) too by virtue of Polya-Szeg¨ o inequality.

rad(RN) → Lq(RN) is com-

h→ z in Lq(RN) as h → ∞ for

h?Lp∩Lp∗(RN)→ 0 we get uh→ z in

h?Lp∩Lp∗(RN)+ ?u∗

h− z?Lq(RN).

(3.3)uh→ uin Lq(RN) as h → ∞, for every p < q < p∗.

In light of (3.3), for a p < q < p∗there exists ζ ∈ Lq(RN), ζ ≥ 0, such that |uh| ≤ ζ for

every h ≥ 1. In turn, by assumption (1.14), for all ε > 0 there exists Cε∈ R with

ε|uh|p+ Cεζq− g(uh)uh≥ 0.

Then, by Fatou’s Lemma, by the arbitrariness of ε and the boundedness of (uh) in Lp(RN),

?

limsup

h

RNg(uh)uh≤

?

RNg(u)u.

Of course, since g(uh)uh≥ 0, again by Fatou’s Lemma one also has

liminf

h

?

RNg(uh)uh≥

?

RNg(u)u,

concluding the proof of formula (3.2)

?

Next, we state the main technical tool for the proof of the second theorem.

Lemma 3.3. Assume that conditions (1.5)-(1.8) and (1.14)-(1.15) hold and that fλsatis-

fies the concrete-(SBPS)cfor all c ∈ R and all λ ∈ [δ0,1]. Then there exists a sequence

(λj,uj) ⊂ [δ0,1] × W1,p(RN) with λjր 1 where ujis a distributional solution of

jt(u,|Du|)Du

|Du|

−div

?

?

+ js(u,|Du|)+ V (|x|)up−1= λjg(u)

in RN,

such that fλj(uj) = cλjand uj= u∗

j.

Proof. The result follows by applying [20, Corollary 3.3] with the following choice of spaces:

X = W1,p(RN), S = W1,p(RN,R+) and V = Lp∩ Lp∗(RN). In fact, it is readily verified

that assumptions (H1)-(H4) in [20, section 3.1] are fulfilled with uH= |u|H, where vH

denotes the standard polarization of v ≥ 0 and with u∗= |u|∗where v∗denotes the

Schwarz symmetrization of v ≥ 0. Condition (H1) is just the continuity of the functionals

fλ. Condition (H2) is satisfied since Lemma 2.1 and Lemma 2.2 hold with the same proof

(notice that the function z in the proof of Lemma 2.1 satisfies z = z∗). Condition (H3)

follows, as in the proof of Lemma 2.5 by a simple direct computation. Assumption (H4)

is satisfied by (1.15) and standard arguments (see also [20, Remark 3.4]). Notice that the

function w = γ(1) = z(x/t0) detected in Lemma 2.1 and used to build the minimax class

Γ is radially symmetric and radially decreasing, so that wH= w for every half space H, as

required in (H4).

?

Page 11

MOUNTAIN PASS SOLUTIONS FOR QUASI-LINEAR EQUATIONS 11

3.1. Proof of Theorem 1.2 concluded. The proof goes along the lines of the proof of

Theorem 1.1 by simple adaptations of the preparatory results contained in Section 2 to the

new setting. With respect to the main differences in the proofs, it is sufficient to replace

Proposition 2.4 with Proposition 3.2 and Lemma 2.5 with Lemma 3.3.

Remark 3.4. In the notations c and cradmentioned at the end of the introduction, we

always have c ≤ crad. On the other hand, when V is constant and the function t ?→ j(s,t)

is p-homogeneous, then c ≥ crad. In fact, let urbe a radial solution at level c provided by

Theorem 1.2, namely f1(ur) = c. Then, defining the radial curve γr(t)(x) := ur(x/tt0),

which belongs to C([0,1],W1,p

proof of Theorem 3.2] through Pohˇ ozaev identity, it follows that

rad(RN)) for a suitable t0> 1 and arguing as in [10, Step I,

c = f1(ur) = max

t∈[0,1]f1(γr(t)),

immediately yielding c ≥ crad, as desired.

Acknowledgment. The authors wish to thank Jean Van Schaftingen for a useful discus-

sion about the comparison between the Mountain Pass levels cradand c.

References

[1] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applica-

tions, J. Functional Anal. 14 (1973), 349–381.

[2] D. Arcoya, L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch.

Rational Mech. Anal. 134 (1996), 249–274.

[3] A. Azzollini, A. Pomponio, On the Schr¨ odinger equation in under the effect of a general nonlinear

term, Indiana Univ. Math. J. 58 (2009), 1361–1378

[4] H. Berestycki, P.L. Lions, Non-linear scalar eld equations. I. Existence of a ground state, Arch.

Rational Mech. Anal., 82 (1983), 313–345.

[5] L. Boccardo, F. Murat, Almost everywhere convergence of the gradients of solutions to elliptic

and parabolic equations, Nonlinear Anal. 19 (1992), 581–597.

[6] A. Canino, M. Degiovanni, Nonsmooth critical point theory and quasilinear elliptic equations.

Topological methods in differential equations and inclusions (Montreal, PQ, 1994), 1–50, NATO Adv.

Sci. Inst. Ser. C Math. Phys. Sci., 472, Kluwer Acad. Publ., Dordrecht, 1995

[7] J.N. Corvellec, M. Degiovanni, M. Marzocchi, Deformation properties for continuous func-

tionals and critical point theory, Topol. Methods. Nonlinear Anal. 1 (1993), 151–171.

[8] M. Degiovanni, M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. Pura

Appl. 167 (1994), 73–100.

[9] M. Degiovanni, A. Musesti, M. Squassina, On the regularity of solutions in the Pucci-Serrin

identity, Calc. Var. Partial Differential Equations 18 (2003), 317–334.

[10] A. Giacomini, M. Squassina Multi-peak solutions for a class of degenerate elliptic equations,

Asymptotic Analysis 36 (2003) 115–147.

[11] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-

Lazer-type problem set on RN, Proc. Roy. Soc. Edinburgh Sect. A 129 (1999), 787–809.

[12] L. Jeanjean, K. Tanaka, A positive solution for a nonlinear Schr¨ odinger equation on RN, Indiana

Univ. Math. J. 54 (2005), 443–464.

[13] L. Jeanjean, J.F. Toland, Bounded Palais-Smale mountain-pass sequences, C.R. Acad. Sci. Paris

S´ er I Math. 327 (1998), 23–28.

Page 12

12 BENEDETTA PELLACCI AND MARCO SQUASSINA

[14] A. Kristaly, V. Radulescu, C.G. Varga, Variational principles in mathematical physics, ge-

ometry, and economics. Qualitative analysis of nonlinear equations and unilateral problems. With a

foreword by Jean Mawhin. Encyclopedia of Math. and its Applications, 136. Cambridge University

Press, 2010. xvi+368 pp.

[15] O. Ladyzenskaya, N. Uraltseva, Linear and quasilinear elliptic equations, Academic Press, 1968.

[16] B. Pellacci, Critical points for non-differentiable functionals, Boll. UMI B 11 (1997), 733–749.

[17] J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302.

[18] M. Squassina, Weak solutions to general Euler’s equations via non-smooth critical point theory,

Ann. Fac. Sci. Toulouse Math. 9 (2000) 113–131.

[19] M. Squassina, On the Palais principle for nonsmooth functionals, preprint, arXiv:1007.3593.

[20] M. Squassina, On Struwe-Jeanjean-Toland monotonicity trick, preprint, arXiv:1010.5410.

[21] M. Squassina, Existence, multiplicity, perturbation, and concentration results for a class of quasi-

linear elliptic problems, Electron. J. Differential Equations Monograph 7 2006, +213, TX, USA.

[22] M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamil-

tonian systems. Fourth edition. Springer-Verlag, Berlin, 2008, +302pp.

[23] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential

Equations 51 (1984), 126–150.

Dipartimento di Scienze Applicate

Universit` a di Napoli Parthenope

Isola C4, I-80143 Napoli, Italy

E-mail address: pellacci@uniparthenope.it

Dipartimento di Informatica

Universit` a degli Studi di Verona

C´ a Vignal 2, Strada Le Grazie 15, I-37134 Verona, Italy

E-mail address: marco.squassina@univr.it