ROBIN BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR ...

Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 100, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

ROBIN BOUNDARY-VALUE PROBLEMS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH SUBCRITICAL AND CRITICAL NONLINEARITIES DIMITRIOS A. KANDILAKIS, MANOLIS MAGIROPOULOS

Abstract. By using variational methods we study the existence of positive solutions for a class of quasilinear elliptic problems with Robin boundary conditions.

1. Introduction Let Ω be a bounded domain in RN with a smooth boundary ∂Ω. In this article we study the nonlinear Robin problem: −∆p u = λ|u|p−2 u + a(x)|u|q−2 u in Ω, |∇u|p−2

∂u + b(x)|u|p−2 u = µρ(x)|u|r−2 u on ∂Ω, ∂η

where ∆p u := div(|∇u|p−2 ∇u), 1 0, a : Ω → R, b, ρ : ∂Ω → R are essentially bounded functions, with b(x) ≥ 0 and mx ∈ ∂Ω : b(·) > 0} > 0. Restrictions on q, r are given in the subsequent sections. With respect to the parameter µ, we notice that its role is crucial in the critical case examined in Section 3. Quasilinear problems of the form −∆p u = f (x, u) with Dirichlet boundary conditions have received considerable attention; see [2, 8, 16, 20, 23]. This equation with Neumann boundary conditions (i.e. b(·) ≡ 0 and ρ(·) ≡ 0) and a(·) being a constant has been studied in [4], where existence of solutions has been provided for λ ∈ (0, λ∗ ), for a suitable λ∗ > 0. The same authors in [3] provide positive solutions to the aforementioned problem but with a critical term added to the right hand side of (1). In [5] the existence of solutions is proved for (1)-(1) when λ appears on the boundary condition, a(·) ≡ 0, and r can be subcritical, critical or supercritical. Multiplicity of solutions is examined in [18] where the right hand side of (1) is a real Carath´eodory function f (x, u, λ) defined on Ω × R × (0, +∞) and the boundary condition is Neumann. Multiplicity of solutions is also proved in [17] for λ > λ2 , 2010 Mathematics Subject Classification. 35J50, 35J65, 47J10. Key words and phrases. Quasilinear elliptic problems; Robin boundary condition; subcritical nonlinearities; critical nonlinearities; fibering method; mountain pass theorem. c

2016 Texas State University. Submitted February 2, 2016. Published April 19, 2016. 1

2

D. A. KANDILAKIS, M. MAGIROPOULOS

EJDE-2016/100

for λ2 being the second eigenvalue of the p-Laplacian operator with Robin boundary conditions, while in [19] existence of positive solutions is shown for λ < λ1 . Existence of solutions depending on the Fu˘cik spectrum of the p-Laplace operator is examined in [24]. When Ω is an exterior domain, existence and nonexistence of solutions is examined in [10]. In case the potential is nonsmooth we refer to [11]. The fibering method, attributed to Pohozaev, is useful when the right hand sides of the equation and the boundary condition are power-like, see [7], [21]. For systems of equations the interested reader may see [6]. Our aim in this work is to provide existence results concerning positive solutions to (1)-(1) when q is either subcritical or critical, r is subcritical and λ ≤ λ1 , where λ1 is the first eigenvalue of the associated eigenvalue problem. When the exponents are subcritical, our proofs rely on the fibering method and the mountain pass theorem developed in Ambrosetti-Rabinowitz [1], while in the case of q being critical we use the concentration-compactness principle of Lions [13, 14]. A useful survey of results concerning the mountain pass theorem is provided in [22]. As usual X := W 1,p (Ω) is equipped with the norm Z Z 1/p kuk1,p = |∇u|p dx + |u|p dx . Ω

Ω

The action functional I(·) corresponding to problem (1)-(1) is defined on X by Z Z Z i 1 µ 1h |∇u|p dx − λ |u|p dx + b(x)|u|p dσ − A(u) − P (u), Iλ (u) = p Ω q r Ω ∂Ω R R r q where P (u) := ∂Ω ρ(x)|u| dσ and A(u) := Ω a(x)|u| dx. Consider the eigenvalue problem − div(|∇u|p−2 ∇u) = λ|u|p−2 u in Ω,

(1.1)

∂u + b(x)|u|p−2 u = 0 on ∂Ω. (1.2) ∂η It is known that the smallest eigenvalue λ1 is isolated and positive with corresponding normalized eigenvector u1 ∈ C 1 (Ω) (that is, ku1 k = 1) which is positive in Ω, [12, Lemma 5.3]. Furthermore, n R |∇u|p dx + R b(x)|u|p dσ o 1,p Ω ∂Ω R λ1 = inf : u ∈ W (Ω)\{0} . (1.3) |u|p dx Ω |∇u|p−2

2. Subcritical exponents In what follows we assume that 1 < q < p∗ :=

Np N −p

and 1 < r < pb∗ :=

p(N −1) N −p .

2.1. Existence of solutions when λ < λ1 . Lemma 2.1. The expression Z Z Z [u] = |∇u|p dx − λ |u|p dx + Ω

Ω

b(x)|u|p dσ

1/p

∂Ω

is a norm on X and is equivalent to k · k1,p . The proof of the above lemma follows from [4, Proposition 2]. Depending on the relative ordering of the exponents p, q, r, we distinguish the following four cases. Case 1. p < min{q, r}. We assume

EJDE-2016/100

ROBIN BOUNDARY-VALUE PROBLEMS

3

(H1) a(·) ≥ 0 and m{x ∈ Ω : a(·) > 0} > 0. (H2) ρ(·) ≥ 0 on ∂Ω and m{x ∈ ∂Ω : ρ(·) > 0} > 0. Let Y be an Banach space and Σ := {A ⊆ X\{0} : A is closed and A = −A}. The genus of a set A ∈ Σ is defined by γ(A) := min{n ∈ N : ∃ϕ ∈ C(A, Rn \{0}) with ϕ(x) = −ϕ(−x)}. Theorem 2.2. Suppose that I : Y → R is an even C 1 (Y, R) function such that: (i) I satisfies the Palais-Smale condition. (ii) I(u) > 0 if 0 < kuk < r and I(u) ≥ c > 0 if kuk = r, for some r > 0. (iii) There exists a subspace Ym ⊆ E of dimension m and a compact subset Am ⊆ Ym with I < 0 on Am such that 0 lies in a bounded component (in Ym ) of Ym \A. Let Γ := {h ∈ C(Y, Y ) : h(0) = 0, h is an odd homeomorhism, I(h(B1 )) ≥ 0}, Km := {K ⊆ Y : K is compact,K = −K, γ(K ∩ h(∂B1 )) ≥ m for every h ∈ Γ}, where B1 denotes the unit ball of Y . Then cm := inf maxI(u) K∈Km u∈K

is a critical value of I with 0 < c < cm ≤ cm+1 < +∞. Furthermore, if cm = cm+1 = · · · = cm+n , then γ(Kcm ) ≥ n + 1, where Kcm := {u ∈ X : I 0 (u) = 0, I(u) = cm }. For the proof of the above Theorem we refer the reader to [1]. Theorem 2.3. Assume that (H1) and (H2) hold. Then (1)-(1) admits infinitely many solutions. Proof. We will show first that I satisfies the Palais-Smale condition. So let {un }n∈N be a sequence in X such that |I(un )| ≤ M and I 0 (un ) → 0. For k ∈ (p, min{q, r}) we have 1 −M + on (1)[un ] ≤ I(un ) − I 0 (un )un ≤ M + on (1)[un ], k and so 1 1 1 1 1 1 −M + on (1)[un ] ≤ − [un ]p + − A(un ) + µ − P (un ) p k k q k r (2.1) ≤ M + on (1)[un ], which implies {un }n∈N is bounded in X. Without loss of generality we may assume that un → u weakly in X and strongly in Lp (Ω), Lq (Ω), Lp (∂Ω) and Lr (∂Ω). Therefore, Z |∇u|p−2 ∇u∇(un − u)dx → 0, (2.2) Ω Z a(|un |q−2 un − |u|q−2 u)(un − u)dx → 0, (2.3) Ω Z b(|un |p−2 un − |u|p−2 u)(un − u)dσ → 0, (2.4) ∂Ω Z ρ(|un |r−2 un − |u|r−2 u)(un − u)dσ → 0 (2.5) ∂Ω

as n → +∞. Since I 0 (un ) → 0, (2.3)-(2.5) imply that hI 0 (un ) − I 0 (u), un − ui → 0

as n → +∞.

(2.6)

4


EJDE-2016/100

Thus, Z

|∇un |p−2 ∇un − |∇u|p−2 ∇u (∇un − ∇u) dx Ω Z − λ (|un |p−2 un − |u|p−2 u)(un − u)dx Z Ω + b(|un |p−2 un − |u|p−2 u)(un − u)dσ Z∂Ω − a(|un |q−2 un − |u|q−2 u)(un − u)dx Ω Z −µ ρ(|un |r−2 un − |u|r−2 u)(un − u)dσ → 0 as n → +∞. ∂Ω

Consequently, Z [|∇un |p−2 ∇un − |∇u|p−2 ∇u](∇un − ∇u)dx → 0

as n → +∞.

Ω

As a consequence of Holder’s inequality we have Z [|∇un |p−2 ∇un − |∇u|p−2 ∇u](∇un − ∇u)dx Ω h Z (p−1)/p Z (p−1)/p i p ≥ |∇un | dx − |∇u|p dx Ω Ω h Z 1/p Z 1/p i p × |∇un | dx − |∇u|p dx . Ω

(2.7)

Ω

Therefore, kun k1,p → kuk1,p . The uniform convexity of X implies that un → u in X. Note that 1 µ 1 1 I(u) = [u]p − A(u) − P (u) ≥ [u]p − c1 [u]q − c2 [u]r , p q r p by the Sobolev embedding, and so I(u) > 0 for kuk = ρ and I(u) ≥ c3 > 0 for kuk < ρ, provided ρ is sufficiently small. Suppose that {Xn }n∈N is a sequence of subspaces of X with dimension dim(Xn ) = n such that ∂u ∂η 6= 0 if u ∈ Xn \{0}. Then, for u ∈ B1n := {v ∈ Xn : [v] = 1} and ζ sufficiently large ζp ζq µζ r ζp ζq µζ r I(ζu) = [u]p − A(u) − P (u) < − minn A(u) − min P (u) < 0. p q r p q u∈B1 r u∈B1n We can now apply Theorem 2.2 to complete the proof. Case 2. 1 < r < q 0. Theorem 2.4. If 1 < r < q 0 or P (u) > 0}. For u ∈ Z, t ≥ 0, one forms tp tq µtr P (u), I(tu) = Hλ (u) − A(u) − p q r where Hλ (u) := [u]p .

EJDE-2016/100


5

For t > 0, let It (tu) = tp−1 Hλ (u) − tq−1 A(u) − µtr−1 P (u). For critical points, we obtain tp Hλ (u) − tq A(u) − µtr P (u) = 0,

(2.8)

that has always a unique solution t = t(u). Let Sλ = Z ∩ {u ∈ X : Hλ (u) = 1}. We notice that {t(u) : u ∈ Sλ } is bounded. b := I(t(u)u). In view of (2.8), For u ∈ Z, we define I(u) b = 1 − 1 t(u)p Hλ (u) + 1 − 1 µt(u)r P (u) < 0. I(u) (2.9) p q q r b is bounded below in Sλ . Let M = inf u∈S I(u). b Notice that I(·) Let {un }n∈N ⊆ Sλ λ b be a minimizing sequence for I/Sλ . Since {un }n∈N is bounded in X, we may assume that un * u in X. At the same time, t(un ) → b t in R. Thus t(un )un * b tu in X. By weak lower semicontinuity of I(·), we have I(b tu) ≤ lim inf I(t(un )un ) = M. n→+∞

Thus b tu 6= 0. Because of the corresponding compact Sobolev embeddings, A(un ) → A(u) and P (un ) → P (u). Exploiting (2.8) for each n, one has t(un )p−r = t(un )q−r A(un ) + µP (un ). Letting n → +∞, we obtain b tp−r = b tq−r A(u) + µP (u).

(2.10)

Since b t > 0, either A(u) > 0 or P (u) > 0, thus u ∈ Z, and t(u) is well defined. The weak lower semicontinuity of the norm applies to give b tp−r Hλ (u) ≤ b tq−r A(u) + µP (u) (2.11) or Hλ (u) ≤ b tq−p A(u) + b tr−p µP (u). At the same time, Hλ (u) = t(u)q−p A(u) + t(u)r−p µP (u). Since the map f (t) = tq−p A(u)+tr−p µP (u), t > 0 is strictly decreasing, the last two relations imply b t ≤ t(u). Let us assume that b t < t(u). We set F (y) := I(yu), y ≥ 0. For y ∈ [b t, t(u)], one has F 0 (y) = y p−1 Hλ (u) − y q−1 A(u) − y r−1 µP (u) = y r−1 [y p−r Hλ (u)−y q−r A(u)−µP (u)], which is negative everywhere but at y = t(u), since (2.8) has a unique solution. Thus F (y) is strictly decreasing in [b t, t(u)], so I(t(u)u) < I(b tu) ≤ M. We take k > 0 such that ku ∈ Sλ (actually, combining (2.10) and (2.11) one sees that k ≥ 1). We have t(ku)p = t(ku)q A(ku) + t(ku)r µP (ku) or

p q r kt(ku) Hλ (u) = kt(ku) A(u) + kt(ku) µP (u), thus kt(ku) = t(u). Then I(t(ku)ku) = I(t(u)u) < I(b tu) ≤ M,

6


EJDE-2016/100

which is a contradiction. Thus b t = t(u), Hλ (u) = 1, and t(u)u is a nontrivial solution of our problem. Since |t(u)u| will also be a minimizer, by Harnack’s inequality we may assume that t(u)u is positive. If a(·) ≤ 0 in Ω, we define Zˆ := {u ∈ X : P (u) > 0}. It is clear that (2.8) has a unique positive solution and M < 0. Furthermore, since the limit u of a minimizing sequence satisfies (2.10), we have that P (u) > 0. Thus u ∈ Zˆ and |t(u)u| is a positive solution of (1),1. Case 3. 1 < q < r < p. Theorem 2.5. Suppose that 1 < q < r < p and (H1), (H2) hold. Then (1)-(1) admits a positive solution. Proof. Note that for every u ∈ Z (2.8) has a unique positive solution t := t(u). b Furthermore, the set {t(u) : u ∈ Z} is bounded. Let I(u) := I(t(u)u). Then, in view of (2.8), b = tp 1 − 1 + µtr 1 − 1 P (u) I(u) p q q r (2.12) 1 1 1 1 1 1 + tp − = tp − < 0. ≤ tp − p q q r p r We can now proceed as in case 2. Case 4. 1 < r 0. Theorem 2.6. If 1 < r < p < q and (H2), (H3) hold, then (1)-(1) admits a positive solution. ˆ Proof. Once more, (2.8) has a unique positive solution t := t(u) for every u ∈ Z. ˆ is bounded and I(u) b ˆ We proceed as Furthermore, the set {t(u) : u ∈ Z} < 0 in Z. in case 2. 2.1.1. Existence of solutions when λ = λ1 . In this section we assume that (H2) and (H3) hold. Case 5. 1 < r < q < p. Theorem 2.7. Assume that 1 < r < q 0 and HλP (u) = b 1}. If u ∈ SλP , then (2.8) has a unique solution t(u) with I(u) < 0. Define b b n ) → M . We claim M = inf u∈SλP I(u) and assume that un ∈ SλP is such that I(u that kun k1,p , n ∈ N, is bounded. Indeed, let us assume that it is not, that is, kun k1,p → +∞. Define zn := udnn , where dn = kun k1,p . Then dpn [zn ]p − dqn A(zn ) = 1. Consequently, [zn ]p ≤ Thus

1 1 → 0, 0 ≤ −A(zn ) ≤ q → 0. dpn dn Z λ1 |zn |p dx → 1. Ω

(2.13) (2.14)

EJDE-2016/100


7

Since kzn k1,p = 1, we may assume that zn → z weakly in X. Therefore, (2.14) implies that Z λ1 |z|p dx = 1, (2.15) Ω

and so z 6= 0. By (2.14) and (2.15) we see that [z] = 0, that is, z is an eigenvector corresponding to λ1 . On the other hand, since A(zn ) → A(z), (2.14) yields A(z) = 0, contradicting the fact z > 0 in Ω. Thus, kun k1,p , n ∈ N, is indeed bounded. So we may assume that un → u weakly in X. Note that, for an infinite number of n0 s, either [un ]p ≥ 21 , or −A(un ) ≥ 12 . In either case, (2.8) implies that r(un ) is bounded. Since (2.8) implies that P (u) > 0, we see that u ∈ SλP . We can now proceed as in case 1 to get a solution. Case 6. 1 < r < p < q. Theorem 2.8. Assume that 1 < r < p < q and (H2), (H3) hold. Then (1)-(1) admits a positive solution. Proof. We use the inequality b = tp 1 − 1 + µtr 1 − 1 P (u) I(u) p q q r 1 1 r 1 r 1 − P (u) + µt − P (u) ≤ µt p q q r 1 1 = µtr − P (u) < 0, p r b to show that inf I(u) < 0 and by following the same steps as in case 5 we obtain P u∈Sλ

a positive solution.

3. The critical case q = p∗

−1) p . with p < r < p(N In this section we study the critical problem q = p∗ := NN−p N −p. and λ < λ1 . The proof follows closely the lines of [9, Theorem 1.4]. Since the ∗ embedding X ,→ Lp (Ω) is no longer compact we do not expect that the PalaisSmale condition holds. So we prove a local Palais-Smale condition which is true if I(·) lies below a certain energy value. In what follows we assume that a(·) ≡ 1 and (H2) holds. Consider the problem

− div(|∇u|p−2 ∇u) = λ|u|p−2 u + |u|q−2 u in Ω, |∇u|p−2 Let

∂u + b(x)|u|p−2 u = µρ(x)h(u) on ∂Ω. ∂η R |∇u|p dx RΩ S= inf , ∗ u∈D 1,p (RN )\{0} Ω |u|p dx

(3.1) (3.2)

be the best Sobolev constant, where u ∈ D1,p (RN ) is the completion of C0∞ (RN ) under the gradient norm. Lemma 3.1. Suppose that {un }n∈N is a sequence in X satisfying the Palais-Smale N condition with energy level c < N1 S p , that is I(un ) → c

and

I 0 (un ) → 0.

Then {un }n∈N has a convergent subsequence in X.

8


EJDE-2016/100

Proof. The boundedness of {un }n∈N is a consequence of (2.1). Thus, {un }n∈N has a subsequence, still denoted by {un }n∈N , which converges weakly to u ∈ X. By [15, Lemma 3.6] there exists a set of points {xj }j∈J ⊆ Ω, J at most countable, and nonnegative numbers µj , νj satisfying X |∇un |p → µ ≥ |∇u|p + µj δxj , j∈J p∗

|un |

→ ν = |u|

p∗

X

+

νj δxj ,

j∈J p ∗

Sνjp ≤ µj p p∗

if xj ∈ Ω,

p N

≤ 2 µj if xj ∈ ∂Ω.

Sνj

Let k ∈ N, ε > 0 and take ϕ ∈ C ∞ (Ω) such that ϕ ≡ 1 in B(xk , ε),

ϕ≡0

in X\B(xk , 2ε),

|∇ϕ| ≤

2 . ε

Since I 0 (un )(ϕun ) → 0 an n → +∞, we obtain Z i hZ |∇un |p−2 ∇un ∇ϕun dx + |∇un |p ϕdx lim n→+∞ Ω Ω Z Z Z Z p p p∗ =λ |u| ϕdx − b(x)|u| ϕdσ + lim |un | ϕdx + µ ρ(x)|u|r ϕdσ n→+∞ Ω Ω ∂Ω ∂Ω Z Z Z Z =λ |u|p ϕdx − |u|p ϕdσ + ϕdν + µ ρ(x)|u|r ϕdσ. Ω

∂Ω

Ω

∂Ω

Note that, by the Holder inequality, Z |∇un |p−2 ∇un ∇ϕun dx lim n→+∞

Ω

≤ lim n→+∞ Z ≤C

Z

p−1 Z 1/p p |un |p ϕdx lim |∇ϕ|p |un |p dx n→+∞

Ω

Ω

1/p |∇ϕ| |u| dx p

p

B(xk ,2ε)∩Ω

≤C

Z

|∇ϕ|N dx

B(xk ,2ε)∩Ω

≤C

0

Z

1/p∗ ∗ |u|p dx

1/N Z B(xk ,2ε)∩Ω

p∗

|u| dx → 0

as ε → 0,

B(xk ,2ε)∩Ω

and so Z Z hZ lim ϕdµ − λ |u|p ϕdx + ε→0

Ω

Ω

b(x)|u|p ϕdσ −

∂Ω

Z

Z ϕdν − µ

Ω

ρ(x)|u|r ϕdσ

i

∂Ω

= µk − νk = 0. p ∗

p ∗

p

Consequently, Sνkp ≤ νk if xk ∈ Ω or 2− N Sνkp ≤ νk if xk ∈ ∂Ω, implying that N N S p ≤ νk if xk ∈ Ω or 12 S p ≤ νk if xk ∈ ∂Ω. On the other hand, c = lim I(un ) = lim I(un ) − lim n→+∞

n→+∞

1

n→+∞ p

I 0 (un )(un )

EJDE-2016/100

= ≥


1 1 − ∗ p p

Z

1 1 − ∗ p p

∗

|u|p +

Ω

Z X

νj δxj + µ

Ω j∈J

1 1 − p r

Z

9

ρ(x)|u|r dσ

∂Ω

1 1 1 − νk = S N/p . p p∗ N

R R ∗ ∗ Thus νk = 0 for every k ∈ J, implying that Ω |un |p dx → Ω |u|p dx. The result follows by exploiting the continuity of the inverse p-Laplace operator. Theorem 3.2. There exists µ0 > 0 such that for µ ≥ µ0 problem (1)-(1) admits a solution. Proof. We will first verify the requirements for the mountain pass theorem. By the Sobolev embedding and trace theorems we see that 1 µ 1 I(u) = [u]p − ∗ A(u) − P (u) p p r ∗ 1 p ≥ [u] − C1 [u]p − C2 [u]r , p for some C1 , C2 > 0,and so for a sufficiently small positive number β there exists a > 0 such that I(u) > a > 0 for [u] = β. We now take v ∈ X\{0}. It is easy to see that lims→+∞ I(sv) = −∞. Thus, I(s0 v) < 0 for sufficiently large s0 . Let c := inf γ∈Γ supt∈[0,1] I(γ(t)), where Γ := {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = N

s0 v}. We will show that c < N1 S p for large enough µ. Take z ∈ X such that kzkp∗ = 1. The maximum value of η → I(ηz), η > 0, is assumed at the point ηµ d satisfying dη I(ηµ z) = 0, that is ∗

∗

∗

ηµp [z]p = ηµp kzkpp∗ + µηµr P (z) = ηµp + µηµr P (z). Therefore,

(3.3)

p

ηµ ≤ [z] p∗ −p , which, in view of (3.3), yields limµ→+∞ ηµ = 0. On the other hand, 1 1 1 1 1 1 I(ηµ z) = ηµp − ∗ [z]p + µηµr ∗ − P (z) ≤ ηµp − ∗ [z]p , p p p r p p implying that limµ→+∞ I(ηµ z) = 0. Thus, for large enough µ, say µ ≥ µ0 , I(ηµ z) < N 1 p N S . By Lemma 3.1, I(·) satisfies the Palais-Smale condition and the mountain pass theorem provides a solution. References [1] A. Ambrosetti, P. H. Rabinowitz; Dual variational methods in critical point theory and applications, J. Funct. Analysis, 14(4) (1973), 349-381. [2] T. Bartsch, Z. Liu; Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Commun. Contemp. Math. vol. 6, No. 2 (2004), 245-258. [3] P. A. Binding, P. Drabek, Y. X. Huang; Existence of multiple solutions of critical quasilinear elliptic Neumann problems, Nonlinear Analysis 42 (2000), 613-629. [4] P. A. Binding, P. Drabek, Y. X. Huang; On Neumann boundary value problems for some quasilinear elliptic equations, Electronic J. Diff. Equations 1997, No. 05 (1997), 1-11. [5] J. Fernandez Bonder, J. D. Rossi; Existence results for the p-Laplacian with nonlinear boundary conditions, J. Math. Analysis Appl. 263 (2001), 195-223. [6] Yu. Bozhkov, E. Mitidieri; Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations 190, no. 1 (2003), 239–267. [7] P. Drabek, S. I. Pohozaev; Positive solutions for the p-Laplacian: application of the fibering method, Proc. Royal Soc. Edinburgh 127A (1997), 703-726.

10


EJDE-2016/100

[8] P. Drabek, P. Tak´ aˇ c; Poincar´ e inequality and Palais-Smale condition for the p-Laplacian, Calc. Var. 29 (2007), 31-58. [9] J. Fernandez-Bonder, J. D. Rossi; Existence results for the p-Laplacian with nonlinear boundary conditions, J. Math. Anal. Appl. 263 (2001), 195-223. [10] R. Filippucci, P. Pucci, V. Radulescu; Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Comm. Partial Differential Equations 33, No. 4-6 (2008) 706–717. [11] L. Gasi´ nski, N. S. Papageorgiou; Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, Vol. 8, Chapman and Hall/CRC, Boca Raton, FL, 2006. [12] A. Lˆ e; Eigenvalue problems for the p-Laplacian, Nonlinear Analysis 64 (2006) 1057-1099. [13] P. L. Lions; The concentration-compactness principle in the calculus of variations. The limit case, part 1, Rev. Mat. Iberoamericana 1, No. 1 (1985), 145-201. [14] P. L. Lions; The concentration-compactness principle in the calculus of variations. The limit case, part 2, Rev. Mat. Iberoamericana 1, No. 2 (1985), 45-121. [15] E. S. de Medeiros; Existˆ encia e concentra¸cao de solu¸c˜ ao para o p-Laplaciano com condi¸c˜ ao de Neumann, Doctoral Dissertation, UNICAMP 2001. [16] D. Motreanu, V. V. Motreanu, N. S. Papageorgiou; Positive solutions and multiple solutions at non-resonance, resonance and near resonance for hemivariational inequalities with pLaplacian, Transactions AMS, 360 No5 (2008), 2527-2545. [17] N. Papageorgiou, V. R˘ adulescu, Multiple solutions with precise sign for nonlinear parametric Robin problems, J. Differential Equations 256, No. 7 (2014), 2449–2479. [18] N. S. Papageorgiou, V. D. R˘ adulescu; Nonlinear parametric Robin problems with combined nonlinearities, Advanced Nonlinear Studies 15 (2015) 715-748. [19] N. Papageorgiou, V. R˘ adulescu; Positive solutions for perturbations of the eigenvalue problem for the Robin p-Laplacian, Ann. Acad. Sci. Fenn. Math. 40, No. 1 (2015) 255–277. [20] K. Perera, E. Silva; On singular p-Laplacian problems, Differential Integral Equations 20, No. 1 (2007) 105-120. [21] S. Pohozaev; Nonlinear variational problems via the fibering method. Sections 5 and 6 by Yu. Bozhkov and E. Mitidieri. Handb. Differ. Equ., Handbook of differential equations: stationary partial differential equations. Vol. V, 49–209, Elsevier/North-Holland, Amsterdam, 2008. [22] P. Pucci, V. R˘ adulescu; The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. 9, No. 3 (2010) 543–584. [23] S. Tiwari; N-Laplacian critical problem with discontinuous nonlinearities, Adv. Nonlinear Anal. 4, No. 2 (2015), 109–121. [24] P. Winkert; Multiplicity results for a class of elliptic problems with nonlinear boundary condition, Comm. Pure and Allpied Analysis, 12, No. 2 (2013), 785-802. [25] E. Zeidler; Nonlinear Functional Analysis and its Applications, vol. I, Springer, NY, 1986. Dimitrios A. Kandilakis School of Architectural Engineering, Technical University of Crete, 73100 Chania, Greece E-mail address: [email protected] Manolis Magiropoulos Department of Electrical Engineering, Technological Educational Institute of Crete, 71410 Heraklion, Crete, Greece E-mail address: [email protected]