Infinitely Many Nodal Solutions for Nonlinear Nonhomogeneous Robin Problems

Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu

doi:10.1515/ans-2015-5040

Article Open Access

Infinitely Many Nodal Solutions for Nonlinear Nonhomogeneous Robin Problems

Nikolaos S. Papageorgiou and Vicenţiu D. Rădulescu

Published/Copyright: April 13, 2016

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From the journal Advanced Nonlinear Studies Volume 16 Issue 2

Abstract

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator which incorporates the p-Laplacian as special case. The reaction f⁢(z,x) is a Carathéodory function which need not satisfy a global growth condition and is only assumed to be odd near zero. Using variational tools, we show that the problem has a whole sequence of distinct nodal (that is, sign-changing) solutions.

Keywords: Nonhomogeneous Differential Operator; Nodal Solution; Nonlinear Regularity Theory; Nonlinear Maximum Principle

MSC 2010: 35J20; 35J60

1 Introduction

Let Ω⊆ℝN be a bounded domain with a C2-boundary ∂⁡Ω. We consider the nonlinear, nonhomogeneous Robin problem

(1.1) { - div ⁡ a ⁢ ( D ⁢ u ⁢ ( z ) ) = f ⁢ ( z , u ⁢ ( z ) ) in ⁢ Ω , ∂ ⁡ u ∂ ⁡ n a + β ⁢ ( z ) ⁢ | u | p - 2 ⁢ u = 0 on ⁢ ∂ ⁡ Ω .

In this problem, a:ℝN→ℝN is a strictly monotone, continuous map which satisfies certain other regularity and growth conditions listed in the hypotheses H⁢(a) below. These conditions are general enough to incorporate in our framework many differential operators of interest, such as the p-Laplace operator, 1<p<∞, and the sum of a p-Laplacian with a q-Laplacian, 1<q<p<∞. The reaction term f⁢(z,x) is a Carathéodory function (that is, for all x∈ℝ, the map z↦f⁢(z,x) is measurable, while, for almost all z∈Ω, the function x↦f⁢(z,x) is continuous).

The interesting feature of our work here is that we do not impose any global growth condition on f⁢(z,⋅). Instead, we assume a local symmetry condition, namely, we require that, for almost all z∈Ω, the function x↦f⁢(z,x) is odd in the bounded interval [-η,η]. In the boundary condition, ∂⁡u∂⁡na denotes the generalized normal derivative corresponding to the differential operator div⁡a⁢(D⁢u) and is defined by

∂ ⁡ u ∂ ⁡ n a = ( a ⁢ ( D ⁢ u ) , n ) ℝ N for all ⁢ u ∈ W 1 , p ⁢ ( Ω )

with n⁢(⋅) being the outward unit normal on ∂⁡Ω. This kind of normal derivative is dictated by the nonlinear Green’s identity (see, for example, Gasiński and Papageorgiou [7, p. 210]) and can be also found in the work of Lieberman [13]. The boundary weight function β∈C0,α⁢(∂⁡Ω) with α∈(0,1) satisfies β⁢(z)≥0 for all z∈∂⁡Ω. When β≡0, then we have the Neumann problem.

Under these general hypotheses on the data of (1.1), we show that there exists a whole sequence {un}n≥1⊆C1⁢(Ω¯) of distinct nodal (that is, sign-changing) solutions. Our approach uses variational tools together with suitable truncation-perturbation techniques. Recently, nodal solutions for nonlinear, nonhomogeneous Robin problems were produced by Papageorgiou and Rădulescu [22, 20]. However, in the aforementioned works, the authors establish the existence of only one nodal solution.

2 Mathematical Background and Hypotheses

Let X be a Banach space and let φ∈C1⁢(X,ℝ). We say that φ satisfies the Palais–Smale condition (PS-condition for short) if every sequence {un}n≥1⊆X such that {φ⁢(un)}n≥1⊆ℝ is bounded and

φ ′ ⁢ ( u n ) → 0 in ⁢ X * ⁢ as ⁢ n → ∞

admits a strongly convergent subsequence.

Our main variational tool will be a variant due to Heinz [10] of a classical result of Clark [4]. The next result is essentially due to Heinz [10] and can be found in Wang [28]. Further extensions with applications to semilinear elliptic Dirichlet problems and to Hamiltonian systems can be found in the works of Liu and Wang [15] and Kajikiya [12].

Theorem 2.1

Let X be a Banach space and assume that φ∈C1⁢(X,ℝ) satisfies the PS-condition, it is even, bounded from below, φ⁢(0)=0 and, for every n∈ℕ, there exist an n-dimensional subspace Yn of X and ρn>0 such that

sup ⁡ { φ ⁢ ( u ) : u ∈ Y n ∩ ∂ ⁡ B ρ n } < 0 ,

where ∂⁡Bρn={u∈X:∥u∥=ρn}. Then, there exists a sequence {un}n≥1 of critical points of φ such that

φ ⁢ ( u n ) < 0 for all ⁢ n ∈ ℕ

and

φ ⁢ ( u n ) → 0 as ⁢ n → ∞ .

Let ϑ∈C1⁢(0,+∞) with ϑ⁢(t)>0 for all t>0 and assume that there exists p>1 such that

(2.1) 0 < c ^ ≤ ϑ ′ ⁢ ( t ) ⁢ t ϑ ⁢ ( t ) ≤ c 0 and c 1 ⁢ t p - 1 ≤ ϑ ⁢ ( t ) ≤ c 2 ⁢ ( 1 + t p - 1 )

for all t>0 and for some c1,c2>0. Then, our hypotheses on the map a⁢(⋅) involved in the definition of the differential operator are that

a⁢(y)=a0⁢(|y|)⁢y for all y∈ℝN with a0⁢(t)>0 for all t>0 and
1. a0∈C1⁢(0,+∞),t↦a0⁢(t)⁢t is strictly increasing on (0,+∞), a0⁢(t)⁢t→0 as t→0+ and
  
  lim t → 0 + ⁡ a 0 ′ ⁢ ( t ) ⁢ t a 0 ⁢ ( t ) > - 1 ;
2. there exists c3>0 such that |∇⁡a⁢(y)|≤c3⁢ϑ⁢(|y|)|y| for all y∈ℝN∖{0};
3. (∇⁡a⁢(y)⁢ξ,ξ)ℝN≥ϑ⁢(|y|)|y|⁢|ξ|2 for all y∈ℝN∖{0} and all ξ∈ℝN;
4. if G0⁢(t)=∫0ta0⁢(s)⁢s⁢𝑑s for t>0, then there exists q∈(1,p) such that
  
  lim sup t → 0 + ⁡ q ⁢ G 0 ⁢ ( t ) t q ≤ c * and t ↦ G 0 ⁢ ( t 1 / q ) ⁢ is convex .

Remark 2.2

Hypotheses H⁢(a) (i)–(iii) come from the nonlinear regularity theory of Lieberman [13] and the nonlinear maximum principle of Pucci and Serrin [26]. Hypothesis H⁢(a) (iv) serves the needs of our problem, but it is a mild condition which is satisfied in all the main cases of interest, as the examples which follow illustrate.

From the above hypotheses it is clear that the primitive G0⁢(⋅) is strictly convex and strictly increasing. We set G⁢(y)=G0⁢(|y|) for all y∈ℝN. Then, G⁢(⋅) is convex, G⁢(0)=0 and

∇ ⁡ G ⁢ ( 0 ) = 0 and ∇ ⁡ G ⁢ ( y ) = G 0 ′ ⁢ ( | y | ) ⁢ y | y | = a 0 ⁢ ( | y | ) ⁢ y = a ⁢ ( y ) for all ⁢ y ∈ ℝ N ∖ { 0 } .

So, G⁢(⋅) is the primitive of a⁢(⋅). The convexity of G⁢(⋅), since G⁢(0)=0, implies that

(2.2) G ⁢ ( y ) ≤ ( a ⁢ ( y ) , y ) ℝ N for all ⁢ y ∈ ℝ N .

The next lemma summarizes the main properties of the map a⁢(⋅). It is a straightforward consequence of hypotheses H⁢(a) (i)–(iii) and of (2.1).

Lemma 2.3

If hypotheses H⁢(a) (i)–(iii) hold, then

y ↦ a ⁢ ( y ) is continuous and strictly monotone, hence, maximal monotone too;
| a ⁢ ( y ) | ≤ c 4 ⁢ ( 1 + | y | p - 1 ) for all y ∈ ℝ N and for some c 4 > 0 ;
(a⁢(y),y)ℝN≥c1p-1⁢|y|p for all y∈ℝN.

The last lemma and (2.2) lead to the following growth estimates for the primitive G⁢(⋅).

Corollary 2.4

If hypotheses H⁢(a) (i)–(iii) hold, then c1p⁢(p-1)⁢|y|p≤G⁢(y)≤c5⁢(1+|y|p) for all y∈ℝN and for some c5>0.

The examples that follow illustrate that our conditions on the map a⁢(⋅) cover many cases of interest.

Example 2.5

The following maps satisfy the hypotheses H⁢(a).

The map a⁢(y)=|y|p-2⁢y with 1<p<∞, which corresponds to the p-Laplacian differential operator defined by

Δ p ⁢ u = div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u ) for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .
The map a⁢(y)=|y|p-2⁢y+|y|q-2⁢y with 1<q<p<∞, which corresponds to the (p,q)-differential operator defined by

Δ p ⁢ u + Δ q ⁢ u for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

Such operators arise in problems of mathematical physics. We mention the works of Benci, D’Avenia, Fortunato and Pisani [1] (quantum physics) and Cherfils and Ilyasov [2] (plasma physics). Recently, existence and multiplicity results for such equations with Dirichlet boundary conditions were proved by Cingolani and Degiovanni [3], Gasiński and Papageorgiou [9], Mugnai and Papageorgiou [17], Papageorgiou and Rădulescu [19, 21, 23] and Sun, Zhang and Su [27].
The map a⁢(y)=(1+|y|2)(p-2)/2⁢y with 1<p<∞, which corresponds to the generalized p-mean curvature differential operator defined by

div ⁡ ( ( 1 + | D ⁢ u | 2 ) ( p - 2 ) / 2 ⁢ D ⁢ u ) for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .
The map a⁢(y)=|y|p-2⁢y⁢(1+11+|y|p) with 1<p<∞, which corresponds to the differential operator

Δ p ⁢ u + div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u 1 + | D ⁢ u | p ) for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) ,

which is used in problems of plasticity.

Finally, we impose the hypothesis that

β∈C0,α⁢(∂⁡Ω) with α∈(0,1) and β⁢(z)≥0 for all z∈∂⁡Ω

and our hypotheses on the reaction term f⁢(z,x) are that

f:Ω×ℝ→ℝ is a Carathéodory function such that, for almost all z∈Ω, f⁢(z,0)=0, f⁢(z,⋅) is odd on [-η,η] for some η>0 with f⁢(z,η)≤0≤f⁢(z,-η) and
1. there exists aη∈L∞⁢(Ω)+ such that |f⁢(z,x)|≤aη⁢(z) for almost all z∈Ω and all |x|≤η;
2. if q∈(1,p) is as in H⁢(a) (iv), then we have
  
  lim x → 0 ⁡ f ⁢ ( z , x ) | x | q - 2 ⁢ x = + ∞ uniformly for almost all ⁢ z ∈ Ω .

Remark 2.6

We stress that the above hypotheses do not impose any global growth condition on f⁢(z,⋅). Instead, we assume that f⁢(z,⋅) has a kind of oscillatory behavior near zero and that it is symmetric in that interval. Hypothesis H⁢(f) (ii) implies the presence of a “concave” term near zero. We mention the work of Liu and Wang [14] who produced infinitely many nodal solutions for a semilinear Schrödinger equation without assuming the existence of zeros. We should point out that the idea of using cut-off techniques to produce an infinity of solutions converging to zero goes back to the work of Wang [28] who modified the reaction term in the interval [-η,η] and applied the result of Clark and Heinz to the modified functional (see Wang [28, Lemma 2.3]).

Using hypothesis H⁢(f) (ii), we see that, given any ξ>0 and recalling that q<p, we can find δ=δ⁢(ξ)∈(0,η^) with η^=min⁡{1,η} such that

(2.3) f ⁢ ( z , x ) ⁢ x ≥ ξ ⁢ | x | q ≥ ξ ⁢ | x | p for almost all ⁢ z ∈ Ω ⁢ and all ⁢ | x | ≤ δ .

Then, given r∈(p,+∞), we can find c6=c6⁢(r,δ)>0 such that

(2.4) f ⁢ ( z , x ) ⁢ x ≥ ξ ⁢ | x | q - c 6 ⁢ | x | r for almost all ⁢ z ∈ Ω ⁢ and all ⁢ x ∈ [ - η , η ] .

From (2.3) we have

(2.5) F ⁢ ( z , x ) ≥ ξ q ⁢ | x | q for almost all ⁢ z ∈ Ω ⁢ and all ⁢ | x | ≤ δ .

In our analysis of (1.1), in addition to the Sobolev space W1,p⁢(Ω), we will also use the Banach space C1⁢(Ω¯). This is an ordered Banach space with positive cone

C + = { u ∈ C 1 ⁢ ( Ω ¯ ) : u ⁢ ( z ) ≥ 0 ⁢ for all ⁢ z ∈ Ω ¯ } .

This cone has a nonempty interior and, if u∈C+ with u⁢(z)>0 for all z∈Ω¯, then u∈int⁢C+. On ∂⁡Ω, we consider the (N-1)-dimensional Hausdorff (surface) measure σ⁢(⋅). Using this measure, we can define the “boundary” Lebesgue spaces Lq⁢(∂⁡Ω), 1≤q≤∞. From the theory of Sobolev spaces we know that there exists a unique continuous linear map γ0:W1,p⁢(Ω)→Lp⁢(∂⁡Ω), known as the “trace map”, such that γ0⁢(u)=u|∂⁡Ω for all u∈W1,p⁢(Ω)∩C⁢(Ω¯). We have

Im γ 0 = W 1 / p ′ , p ( ∂ Ω ) ( 1 p + 1 p ′ = 1 ) and ker γ 0 = W 0 1 , p ( Ω ) .

Moreover, the trace map γ0 is compact into Lq⁢(∂⁡Ω) for q∈[1,(N-1)⁢pN-p). Hereafter, for the sake of notational simplicity, we will drop the use of the trace map γ0. The restrictions of Sobolev functions on ∂⁡Ω are understood in the sense of traces. By ∥⋅∥ we denote the norm of the Sobolev space W1,p⁢(Ω) defined by

∥ u ∥ = [ ∥ u ∥ p p + ∥ D ⁢ u ∥ p p ] 1 / p for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

For every x∈ℝ, let x±=max⁡{±x,0}. Then, for u∈W1,p⁢(Ω), we set u±⁢(⋅)=u⁢(⋅)±. We know that

u = u + - u - , | u | = u + + u - and u + , u - ∈ W 1 , p ⁢ ( Ω ) .

Also, by |⋅|N we denote the Lebesgue measure on ℝN and A:W1,p⁢(Ω)→W1,p⁢(Ω)* is the nonlinear map defined by

〈 A ⁢ ( u ) , h 〉 = ∫ Ω ( a ⁢ ( D ⁢ u ) , D ⁢ h ) ℝ N ⁢ 𝑑 z for all ⁢ u , h ∈ W 1 , p ⁢ ( Ω ) .

The map A:W1,p⁢(Ω)→W1,p⁢(Ω)* is continuous, monotone and of type (S)+, that is,

u n → w u ⁢ in ⁢ W 1 , p ⁢ ( Ω ) and lim sup n → ∞ ⁡ 〈 A ⁢ ( u n ) , u n - u 〉 ≤ 0 ⁢ implies that ⁢ u n → u ⁢ in ⁢ W 1 , p ⁢ ( Ω ) .

Here, by 〈⋅,⋅〉 we denote the duality brackets for the pair (W1,p⁢(Ω)*,W1,p⁢(Ω)) (see Gasiński and Papageorgiou [8]). Finally, for any φ∈C1⁢(X,ℝ), by Kφ we denote the critical set of φ, that is,

K φ = { u ∈ X : φ ′ ⁢ ( u ) = 0 } .

3 Nodal Solutions

Using (2.4), we introduce the truncation

(3.1) e ( z , x ) = { - ξ ⁢ η q - 1 + c 6 ⁢ η r - 1 if ⁢ x < - η , ξ ⁢ | x | q - 2 ⁢ x - c 6 ⁢ | x | r - 2 ⁢ x if - η ≤ x ≤ η , ξ ⁢ η q - 1 - c 6 ⁢ η r - 1 if ⁢ η < x

of the right-hand side of (2.4) and, for all (z,x)∈∂⁡Ω×ℝ, the truncation

(3.2) b ( z , x ) = { - β ⁢ ( z ) ⁢ η p - 1 if ⁢ x < - η , β ⁢ ( z ) ⁢ | x | p - 2 ⁢ x if - η ≤ x ≤ η , β ⁢ ( z ) ⁢ η p - 1 if ⁢ η < x

of the boundary term β⁢(z)⁢|x|p-2⁢x. Both are Carathéodory functions. We consider the auxiliary nonlinear, nonhomogeneous Robin problem

(3.3) { - div ⁡ a ⁢ ( D ⁢ u ⁢ ( z ) ) = e ⁢ ( z , u ⁢ ( z ) ) in ⁢ Ω , ∂ ⁡ u ∂ ⁡ n a + b ⁢ ( z , u ) = 0 on ⁢ ∂ ⁡ Ω .

Proposition 3.1

If hypotheses H⁢(a), H⁢(β) and H⁢(f) hold, then (3.3) admits a unique positive solution u¯∈int⁢C+ and v¯=-u¯∈-int⁢C+ is its unique negative solution.

Proof.

We introduce the Carathéodory function τ:Ω×ℝ→ℝ defined by

(3.4) τ ( z , x ) = { - η p - 1 if ⁢ x < - η , | x | p - 2 ⁢ x if - η ≤ x ≤ η , η p - 1 if ⁢ η < x .

Let

T ⁢ ( z , x ) = ∫ 0 x τ ⁢ ( z , s ) ⁢ 𝑑 s , E ⁢ ( z , x ) = ∫ 0 x e ⁢ ( z , s ) ⁢ 𝑑 s and B ⁢ ( z , x ) = ∫ 0 x b ⁢ ( z , s ) ⁢ 𝑑 s ,

and consider the C1-functional ψ+:W1,p⁢(Ω)→ℝ defined by

ψ + ⁢ ( u ) = ∫ Ω G ⁢ ( D ⁢ u ) ⁢ 𝑑 z + 1 p ⁢ ∥ u ∥ p p + ∫ ∂ ⁡ Ω B ⁢ ( z , u + ) ⁢ 𝑑 σ - ∫ Ω E ⁢ ( z , u + ) ⁢ 𝑑 z - ∫ Ω T ⁢ ( z , u + ) ⁢ 𝑑 z for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

From Corollary 2.4 and (3.1), (3.2) and (3.4) it is clear that ψ+ is coercive. Also, using the Sobolev embedding theorem and the trace theorem, we see that ψ+ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find u¯∈W1,p⁢(Ω) such that

(3.5) ψ + ⁢ ( u ¯ ) = inf ⁡ { ψ + ⁢ ( u ) : u ∈ W 1 , p ⁢ ( Ω ) } .

Hypothesis H⁢(a) (iv) implies that we can find c1*>c* and δ1∈(0,η^) such that

(3.6) G 0 ⁢ ( t ) ≤ c 1 * q ⁢ t q for all ⁢ t ∈ [ 0 , δ 1 ] .

Let u∈int⁢C+ and choose t∈(0,1) small such that

(3.7) t ⁢ u ⁢ ( z ) ≤ δ 1 and t ⁢ | D ⁢ u ⁢ ( z ) | ≤ δ 1 for all ⁢ z ∈ Ω ¯ .

Using (3.6), (3.7), (3.1), (3.2) and (3.4), we have (see hypothesis H⁢(β) and the trace theorem)

ψ + ⁢ ( t ⁢ u ) ≤ t p ⁢ c 1 * p ⁢ ∥ D ⁢ u ∥ p p + t p p ⁢ ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u p ⁢ 𝑑 σ - t q ⁢ ξ q ⁢ ∥ u ∥ q q + t r ⁢ c 6 r ⁢ ∥ u ∥ r r

≤ ( t p - q ⁢ c 1 * p ⁢ ∥ D ⁢ u ∥ p p + t p - q p ⁢ c 8 ⁢ ∥ u ∥ p - ξ q ⁢ ∥ u ∥ q q + t r - q r ⁢ c 8 ⁢ ∥ u ∥ r r ) ⁢ t q

for some c8>0. Since 1<q<p<r, choosing t∈(0,1) even smaller if necessary, we have that ψ+⁢(t⁢u)<0 implies (see (3.5))

ψ + ⁢ ( u ¯ ) < 0 = ψ + ⁢ ( 0 )

and, hence, u¯≠0. From (3.5) we have that ψ+′⁢(u¯)=0 implies

(3.8) 〈 A ⁢ ( u ¯ ) , h 〉 + ∫ Ω | u ¯ | p - 2 ⁢ u ¯ ⁢ h ⁢ 𝑑 z + ∫ ∂ ⁡ Ω b ⁢ ( z , u + ) ⁢ h ⁢ 𝑑 σ = ∫ Ω ( e ⁢ ( z , u + ) + τ ⁢ ( z , u + ) ) ⁢ h ⁢ 𝑑 z for all h ∈ W 1 , p ⁢ ( Ω ) .

In (3.8), first we choose h=-u¯-∈W1,p⁢(Ω) and then we have that (see Lemma 2.3 and (3.1), (3.2) and (3.4))

c 1 p - 1 ⁢ ∥ D ⁢ u ¯ - ∥ p p + ∥ u - ∥ p p ≤ 0

implies u¯≥0 and u¯≠0. Also, in (3.8), we choose h=(u¯-η)+∈W1,p⁢(Ω) and then we have (see (3.1), (3.2) and (3.4) for the equality and (2.4) for the first inequality)

〈 A ⁢ ( u ¯ ) , ( u ¯ - η ) + 〉 + ∫ Ω u ¯ p - 1 ⁢ ( u ¯ - η ) + ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ η p - 1 ⁢ ( u ¯ - η ) + ⁢ 𝑑 σ

= ∫ Ω ( ξ ⁢ η q - 1 - c 6 ⁢ η r - 1 + η p - 1 ) ⁢ ( u ¯ - η ) + ⁢ 𝑑 z

≤ ∫ Ω ( f ⁢ ( z , η ) + η p - 1 ) ⁢ ( u ¯ - η ) + ⁢ 𝑑 z

≤ 〈 A ⁢ ( η ) , ( u ¯ - η ) + 〉 + ∫ Ω η p - 1 ⁢ ( u ¯ - η ) + ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ η p - 1 ⁢ ( u ¯ - η ) + ⁢ 𝑑 σ

since A⁢(η)=0 and f⁢(z,η)≤0 for almost all z∈Ω, which implies that

〈 A ⁢ ( u ¯ ) - A ⁢ ( η ) , ( u ¯ - η ) + 〉 + ∫ Ω ( u ¯ p - 1 - η p - 1 ) ⁢ ( u ¯ - η ) + ⁢ 𝑑 z ≤ 0 .

Therefore,

| { u ¯ > η } | N = 0 ,

that is,

u ¯ ≤ η .

Thus, we have proved that

(3.9) u ¯ ∈ [ 0 , η ] = { u ∈ W 1 , p ⁢ ( Ω ) : 0 ≤ u ⁢ ( z ) ≤ η ⁢ for almost all ⁢ z ∈ Ω } and u ¯ ≠ 0 .

Then, using (3.1), (3.2), (3.4) and (3.9), we see that (3.8) becomes

〈 A ⁢ ( u ¯ ) , h 〉 + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u ¯ p - 1 ⁢ h ⁢ 𝑑 σ = ∫ Ω e ⁢ ( z , u ¯ ) ⁢ h ⁢ 𝑑 z for all ⁢ h ∈ W 1 , p ⁢ ( Ω ) ,

which gives (see Papageorgiou and Rădulescu [18])

{ - div ⁡ a ⁢ ( D ⁢ u ¯ ⁢ ( z ) ) = e ⁢ ( z , u ¯ ⁢ ( z ) ) for almost all ⁢ z ∈ Ω , ∂ ⁡ u ¯ ∂ ⁡ n a + β ⁢ ( z ) ⁢ u ¯ p - 1 = 0 on ⁢ ∂ ⁡ Ω ,

that is, u¯ is a positive solution of (3.3). From Papageorgiou and Rădulescu [24] we have that u¯∈L∞⁢(Ω) and, then, the nonlinear regularity result of Lieberman [13, p. 320]) implies that u¯∈C+∖{0}. Because of (3.9) we have

- div ⁡ a ⁢ ( D ⁢ u ¯ ⁢ ( z ) ) = ξ ⁢ u ¯ ⁢ ( z ) q - 1 - c 6 ⁢ u ¯ ⁢ ( z ) r - 1 for almost all ⁢ z ∈ Ω ,

which gives

div ⁡ a ⁢ ( D ⁢ u ¯ ⁢ ( z ) ) ≤ c 6 ⁢ η r - p ⁢ u ¯ ⁢ ( z ) p - 1 for almost all ⁢ z ∈ Ω ,

that is (see Pucci and Serrin [26, pp. 111, 120]),

u ¯ ∈ int ⁢ C + .

Next, we show the uniqueness of this positive solution. To this end, we consider the integral functional j:L1⁢(Ω)→ℝ¯=ℝ∪{+∞} defined by

j ( u ) = { ∫ Ω G ⁢ ( D ⁢ u 1 / q ) ⁢ 𝑑 z + 1 p ⁢ ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u p / q ⁢ 𝑑 σ if ⁢ u ≥ 0 , u 1 / q ∈ W 1 , p ⁢ ( Ω ) , + ∞ otherwise .

Let u1,u2∈dom⁡j={u∈L1⁢(Ω):j⁢(u)<+∞} (the effective domain of j ). We set

u = ( ( 1 - t ) ⁢ u 1 + t ⁢ u 2 ) 1 / q for ⁢ t ∈ [ 0 , 1 ] .

Using Díaz and Saá [5, Lemma 1], we have

| D ⁢ u ⁢ ( z ) | ≤ [ ( 1 - t ) ⁢ | D ⁢ u 1 ⁢ ( z ) 1 / q | q + t ⁢ | D ⁢ u 2 ⁢ ( z ) 1 / q | q ] 1 / q for almost all ⁢ z ∈ Ω

and because G0⁢(⋅) is increasing and from hypothesis H⁢(a) (iv), for almost all z∈Ω, we have

G 0 ⁢ ( | D ⁢ u ⁢ ( z ) | ) ≤ G 0 ⁢ ( [ ( 1 - t ) ⁢ | D ⁢ u 1 ⁢ ( z ) 1 / q | q + t ⁢ | D ⁢ u 2 ⁢ ( z ) 1 / q | q ] 1 / q ) ≤ ( 1 - t ) ⁢ G 0 ⁢ ( | D ⁢ u 1 ⁢ ( z ) 1 / q | ) + t ⁢ G 0 ⁢ ( | D ⁢ u 2 ⁢ ( z ) 1 / q | ) ,

which gives

G ⁢ ( D ⁢ u ⁢ ( z ) ) ≤ ( 1 - t ) ⁢ G ⁢ ( D ⁢ u 1 ⁢ ( z ) 1 / q ) + t ⁢ G ⁢ ( D ⁢ u 2 ⁢ ( z ) 1 / q ) for almost all ⁢ z ∈ Ω ,

that is, j⁢(⋅) is convex (recall that q<p and see hypothesis H⁢(β)) By Fatou’s lemma, j⁢(⋅) is lower semicontinuous.

Let y¯∈W1,p⁢(Ω) be another positive solution of (3.3). As we did for u¯ in the first part of the proof, we can show that

y ¯ ∈ [ 0 , η ] ∩ int ⁡ C + .

For any h∈C1⁢(Ω¯) and for |t|<1 small, we have

u ¯ q + t ⁢ h ∈ dom ⁡ j and y ¯ q + t ⁢ h ∈ dom ⁡ j .

Then, we see that the functional j⁢(⋅) is Gâteaux differentiable at u¯q and y¯q in the direction h. Moreover, via the chain rule and the nonlinear Green’s identity, we have

j ′ ⁢ ( u ¯ q ) ⁢ ( h ) = 1 q ⁢ ∫ Ω - div ⁡ a ⁢ ( D ⁢ u ¯ ) u ¯ q - 1 ⁢ h ⁢ 𝑑 z and j ′ ⁢ ( y ¯ q ) ⁢ ( h ) = 1 q ⁢ ∫ Ω - div ⁡ a ⁢ ( D ⁢ y ¯ ) y ¯ q - 1 ⁢ h ⁢ 𝑑 z .

Choose h=u¯q-y¯q. Since j⁢(⋅) is convex, j′⁢(⋅) is monotone, and so we have (see (3.1))

0 ≤ ∫ Ω ( - div ⁡ a ⁢ ( D ⁢ u ¯ ) u ¯ q - 1 - - div ⁡ a ⁢ ( D ⁢ y ¯ ) y ¯ q - 1 ) ⁢ ( u ¯ q - y ¯ q ) ⁢ 𝑑 z = ∫ Ω c 6 ⁢ ( y ¯ r - q - u ¯ r - q ) ⁢ ( u ¯ q - y ¯ q ) ⁢ 𝑑 z ,

which gives

u ¯ = y ¯

and, then, u¯∈[0,η]∩int⁢C+ is the unique positive solution of (3.3). Evidently, since x↦ξ⁢|x|q-2⁢x-c6⁢|x|r-2⁢x is odd, we have that v¯=-u¯∈[-η,0]∩(-int⁢C+) is the unique negative solution of (3.3). ∎

We introduce the sets

S + = { u ∈ W 1 , p ⁢ ( Ω ) : u ⁢ is a positive solution of (1.1) with ⁢ u ∈ [ 0 , η ] } ,

S - = { v ∈ W 1 , p ⁢ ( Ω ) : v ⁢ is a negative solution of (1.1) with ⁢ v ∈ [ - η , 0 ] } .

As before, the nonlinear maximum principle implies that

S + ⊆ int ⁡ C + and S - ⊆ - int ⁡ C + .

Moreover, as in Filippakis and Papageorgiou [6], we have that

S + ⁢ is downward directed,

that is, if u1,u2∈S+, then we can find u∈S+ such that u≤min⁡{u1,u2}, and

S - ⁢ is upward directed,

that is, if v1,v2∈S-, then we can find v∈S- such that v≥max⁡{v1,v2} (see also Motreanu, Motreanu and Papageorgiou [16, p. 421]).

Proposition 3.2

If hypotheses H⁢(a), H⁢(β) and H⁢(f) hold, then u¯≤u for all u∈S+ and v≤v¯ for all v∈S-.

Proof.

Let u∈S+. We consider the Carathéodory functions k+⁢(z,x),b^+⁢(z,x) and τ^+⁢(z,x) defined by

(3.10) k + ( z , x ) = { 0 if ⁢ x < 0 , e ⁢ ( z , x ) if ⁢ 0 ≤ x ≤ u ⁢ ( z ) , e ⁢ ( z , u ⁢ ( z ) ) if ⁢ u ⁢ ( z ) < x ,

(3.11) b ^ + ( z , x ) = { 0 if ⁢ x < 0 , β ⁢ ( z ) ⁢ x p - 1 if ⁢ 0 ≤ x ≤ u ⁢ ( z ) , β ⁢ ( z ) ⁢ u ⁢ ( z ) p - 1 if ⁢ u ⁢ ( z ) < x , for all ( z , x ) ∈ ∂ Ω × ℝ ,

(3.12) τ ^ + ( z , x ) = { 0 if ⁢ x < 0 , x p - 1 if ⁢ 0 ≤ x ≤ u ⁢ ( z ) , u ⁢ ( z ) p - 1 if ⁢ u ⁢ ( z ) < x .

We set

K + ⁢ ( z , x ) = ∫ 0 x k + ⁢ ( z , s ) ⁢ 𝑑 s , B ^ + ⁢ ( z , x ) = ∫ 0 x b ^ + ⁢ ( d , s ) ⁢ 𝑑 s and T ^ + ⁢ ( z , x ) = ∫ 0 x τ ^ + ⁢ ( z , s ) ⁢ 𝑑 s .

Consider the C1-functional γ+:W1,p⁢(Ω)→ℝ defined by

γ + ⁢ ( u ) = ∫ Ω G ⁢ ( D ⁢ u ) ⁢ 𝑑 z + 1 p ⁢ ∥ u ∥ p p + ∫ ∂ ⁡ Ω B ^ + ⁢ ( z , u ) ⁢ 𝑑 σ - ∫ Ω K + ⁢ ( z , u ) ⁢ 𝑑 z - ∫ Ω T ^ + ⁢ ( z , u ) ⁢ 𝑑 z for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

From Corollary 2.4 and (3.10), (3.11) and (3.12) we see that γ+ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u¯0∈W1,p⁢(Ω) such that

(3.13) γ + ⁢ ( u ¯ 0 ) = inf ⁡ { γ + ⁢ ( u ) : u ∈ W 1 , p ⁢ ( Ω ) } .

As before (see the proof of Proposition 3.1), since 1<q<p<r, for u~∈int⁢C+ and t∈(0,1) small, we have (see hypothesis H⁢(a) (iv))

γ + ⁢ ( t ⁢ u ~ ) < 0 = γ + ⁢ ( 0 ) ,

which implies (see (3.13))

γ + ⁢ ( u ¯ 0 ) < 0 = γ + ⁢ ( 0 )

and, hence, u¯0≠0. From (3.13) we have that γ+′⁢(u¯0)=0 implies

(3.14) 〈 A ⁢ ( u ¯ 0 ) , h 〉 + ∫ Ω | u ¯ 0 | p - 2 ⁢ u ¯ 0 ⁢ h ⁢ 𝑑 z + ∫ ∂ ⁡ Ω b ^ + ⁢ ( z , u ¯ 0 ) ⁢ h ⁢ 𝑑 z = ∫ Ω ( k + ⁢ ( z , u ¯ 0 ) + τ ^ + ⁢ ( z , u ¯ 0 ) ) ⁢ h ⁢ 𝑑 z for all ⁢ h ∈ W 1 , p ⁢ ( Ω ) .

In (3.14), we choose h=-u¯0-. Using Lemma 2.3 and (3.10), (3.11) and (3.12), we obtain that

c 1 p - 1 ⁢ ∥ D ⁢ u ¯ 0 - ∥ p p + ∥ u ¯ 0 - ∥ p p ≤ 0

implies u¯0≥0 and u¯0≠0. Also, in (3.14), we choose h=(u¯0-u)+∈W1,p⁢(Ω). Then, we have (see (3.10), (3.11) and (3.12) for the first equality, see (3.1) and recall that u∈[0,η] for the second one and see (2.4) for the first inequality)

〈 A ⁢ ( u ¯ 0 ) , ( u ¯ 0 - u ) + 〉 + ∫ Ω u ¯ 0 p - 1 ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u p - 1 ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 σ

= ∫ Ω ( e ⁢ ( z , u ) + u p - 1 ) ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 z

= ∫ Ω ( ξ ⁢ u q - 1 - c 6 ⁢ u r - 1 + u p - 1 ) ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 z

≤ ∫ Ω ( f ⁢ ( z , u ) + u p - 1 ) ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 z

= 〈 A ⁢ ( u ) , ( u ¯ 0 - u ) + 〉 + ∫ Ω u p - 1 ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u p - 1 ⁢ ( u 0 - u ) + ⁢ 𝑑 σ

since u∈S+, which implies that

〈 A ⁢ ( u ¯ 0 ) - A ⁢ ( u ) , ( u ¯ 0 - u ) + 〉 + ∫ Ω ( u ¯ 0 p - 1 - u p - 1 ) ⁢ ( u ¯ 0 - u ) + ⁢ 𝑑 z ≤ 0 .

Therefore,

| { u ¯ 0 > u } | N = 0 ,

that is,

u ¯ 0 ≤ u .

Thus, we have proved that

(3.15) u ¯ 0 ∈ [ 0 , u ] = { y ∈ W 1 , p ⁢ ( Ω ) : 0 ≤ y ⁢ ( z ) ≤ u ⁢ ( z ) ⁢ for almost all ⁢ z ∈ Ω } and u ¯ 0 ≠ 0 .

Because of (3.10), (3.11), (3.12) and (3.15) we have that (3.14) becomes

〈 A ⁢ ( u ¯ 0 ) , h 〉 + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u ¯ 0 p - 1 ⁢ h ⁢ 𝑑 σ = ∫ Ω e ⁢ ( z , u ¯ 0 ) ⁢ h ⁢ 𝑑 z for all ⁢ h ∈ W 1 , p ⁢ ( Ω ) ,

which implies that u¯0 is a positive solution of (3.3) (see Papageorgiou and Rădulescu [18]). Then, from Proposition 3.1 we have u¯0=u¯ and, as a result, u¯≤u for all u∈S+.

In a similar fashion, we show that v≤v¯ for all v∈S+. ∎

Next, we produce extremal constant-sign solutions, that is, a smallest positive solution and a biggest negative solution.

Proposition 3.3

If hypotheses H⁢(a), H⁢(β) and H⁢(f) hold, then (1.1) admits a smallest positive solution

u * ∈ [ 0 , η ] ∩ int ⁡ C +

and a biggest negative solution

v * ∈ [ - η , 0 ] ∩ ( - int ⁡ C + ) .

Proof.

Evidently, we restrict ourselves to the sets S+ and S-. From Hu and Papageorgiou [11, Lemma 3.10] we know that we can find a decreasing sequence {un}n≥1⊆S+ such that

inf ⁡ S + = inf n ≥ 1 ⁡ u n .

For all n∈ℕ, we have

(3.16) 〈 A ⁢ ( u n ) , h 〉 + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u n p - 1 ⁢ h ⁢ 𝑑 z = ∫ Ω f ⁢ ( z , u n ) ⁢ h ⁢ 𝑑 z for all ⁢ h ∈ W 1 , p ⁢ ( Ω ) .

Clearly, {un}n≥1⊆W1,p⁢(Ω) is bounded and so we may assume that

(3.17) u n → w u * ⁢ in ⁢ W 1 , p ⁢ ( Ω ) and u n → u * ⁢ in ⁢ L p ⁢ ( Ω ) ⁢ and in ⁢ L p ⁢ ( ∂ ⁡ Ω ) .

In (3.16), we choose h=un-u*∈W1,p⁢(Ω), we pass to the limit as n→∞ and we use (3.17). Then, we have that

lim n → ∞ ⁡ 〈 A ⁢ ( u n ) , u n - u * 〉 = 0

implies

(3.18) u n → u * in ⁢ W 1 , p ⁢ ( Ω )

since A⁢(⋅) is of type (S)+. So, if in (3.16) we pass to the limit as n→∞ and use (3.18), then

(3.19) 〈 A ⁢ ( u * ) , h 〉 + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u * p - 1 ⁢ h ⁢ 𝑑 σ = ∫ Ω f ⁢ ( z , u * ) ⁢ h ⁢ 𝑑 z for all ⁢ h ∈ W 1 , p ⁢ ( Ω ) .

Also (see Proposition 3.2),

(3.20) u ¯ ≤ u * .

From (3.19) and (3.20) we infer that

u * ∈ S + ⊆ int ⁡ C + and u * = inf ⁡ S + .

Similarly, we produce

v * ∈ S - and v * = sup ⁡ S - . ∎

Using these two extremal constant-sign solutions, we introduce the Carathéodory functions

(3.21) μ ( z , x ) = { f ⁢ ( z , v * ⁢ ( z ) ) + | v * ⁢ ( z ) | p - 2 ⁢ v * ⁢ ( z ) if ⁢ x < v * ⁢ ( z ) , f ⁢ ( z , x ) + | x | p - 2 ⁢ x if ⁢ v * ⁢ ( z ) ≤ x ≤ u * ⁢ ( z ) , f ⁢ ( z , u * ⁢ ( z ) ) + u * ⁢ ( z ) p - 1 if ⁢ u * ⁢ ( z ) < x ,

(3.22) b ¯ ( z , x ) = { β ⁢ ( z ) ⁢ | v * ⁢ ( z ) | p - 2 ⁢ v * ⁢ ( z ) if ⁢ x < v * ⁢ ( z ) , β ⁢ ( z ) ⁢ | x | p - 2 ⁢ x if ⁢ v * ⁢ ( z ) ≤ x ≤ u * ⁢ ( z ) , β ⁢ ( z ) ⁢ u * ⁢ ( z ) p - 1 if ⁢ u * ⁢ ( z ) < x .

We set

M ⁢ ( z , x ) = ∫ 0 x μ ⁢ ( z , s ) ⁢ 𝑑 s and B ¯ ⁢ ( z , x ) = ∫ 0 x b ¯ ⁢ ( z , s ) ⁢ 𝑑 s ,

and we consider the C1-functional φ¯:W1,p⁢(Ω)→ℝ defined by

φ ¯ ⁢ ( u ) = ∫ Ω G ⁢ ( D ⁢ u ) ⁢ 𝑑 z + 1 p ⁢ ∥ u ∥ p p + ∫ ∂ ⁡ Ω B ¯ ⁢ ( z , u ) ⁢ 𝑑 σ - ∫ Ω M ⁢ ( z , u ) ⁢ 𝑑 z for all ⁢ u ∈ W 1 , p ⁢ ( Ω ) .

Proposition 3.4

If hypotheses H⁢(a), H⁢(β) and H⁢(f) hold, then φ¯ satisfies the PS-condition, it is even, bounded from below, φ¯⁢(0)=0 and Kφ¯⊆[v*,u*].

Proof.

From (3.21) and (3.22) it is clear that φ¯ is coercive. So, it is bounded from below and satisfies the PS-condition (see Papageorgiou and Winkert [25]). Hypotheses H⁢(f) imply that φ¯ is even (recall that u*∈S+ and v*∈S-) and φ¯⁢(0)=0. Finally, let u∈Kφ¯. Then, φ¯′⁢(u)=0 implies that

(3.23) 〈 A ⁢ ( u ) , h 〉 + ∫ Ω | u | p - 2 ⁢ u ⁢ h ⁢ 𝑑 z + ∫ ∂ ⁡ Ω b ¯ ⁢ ( z , u ) ⁢ h ⁢ 𝑑 σ = ∫ Ω μ ⁢ ( z , u ) ⁢ h ⁢ 𝑑 z for all ⁢ h ∈ W 1 , p ⁢ ( Ω ) .

In (3.23), we first choose h=(u-u*)+∈W1,p⁢(Ω). Then, we have (see (3.21) and (3.22) for the first equality and recall that u*∈S+ for the second one)

〈 A ⁢ ( u ) , ( u - u * ) + 〉 + ∫ Ω u p - 1 ⁢ ( u - u * ) + ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u * p - 1 ⁢ ( u - u * ) + ⁢ 𝑑 σ

= ∫ Ω ( f ⁢ ( z , u * ) + u * p - 1 ) ⁢ ( u - u * ) + ⁢ 𝑑 z

= 〈 A ⁢ ( u * ) , ( u - u * ) + 〉 + ∫ Ω u * p - 1 ⁢ ( u - u * ) + ⁢ 𝑑 z + ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ u * p - 1 ⁢ ( u - u * ) + ⁢ 𝑑 σ ,

which implies that

〈 A ⁢ ( u ) - A ⁢ ( u * ) , ( u - u * ) + 〉 + ∫ Ω ( u p - 1 - u * p - 1 ) ⁢ ( u - u * ) + ⁢ 𝑑 z = 0 .

Therefore,

| { u > u * } | N = 0 ,

that is,

u ≤ u * .

Similarly, if in (3.23) we choose h=(v*-u)+∈W1,p⁢(Ω), then we obtain that v*≤u implies

K φ ¯ ⊆ [ v * , u * ] . ∎

The extremality of v*∈-int⁡C+ and of u*∈int⁡C+ implies the following property.

Corollary 3.5

If hypotheses H⁢(a), H⁢(β) and H⁢(f) hold, then the elements of Kφ¯∖{0,v*,u*} are nodal solutions of (1.1).

Now, we are ready to produce a whole sequence of distinct nodal solutions for (1.1).

Theorem 3.6

Assume that hypotheses H⁢(a), H⁢(β) and H⁢(f) hold. Then, (1.1) has a whole sequence {un}n≥1⊆C1⁢(Ω¯) of distinct nodal solutions.

Proof.

Let η¯=min⁡{minΩ¯⁡u*,-maxΩ¯⁡v*} (recall that u*∈int⁡C+ and v*∈-int⁡C+). Hypothesis H⁢(a) (iv) implies that we can find δ0∈(0,η¯] such that

(3.24) G ⁢ ( y ) ≤ c 9 ⁢ | y | q for all ⁢ y ∈ ℝ N ⁢ with ⁢ | y | ≤ δ 0 ⁢ and for some ⁢ c 9 > 0 .

Also, from (2.5) we have

(3.25) F ⁢ ( z , x ) ≥ ξ q ⁢ | x | q for almost all ⁢ z ∈ Ω ⁢ and for all ⁢ | x | ≤ δ ⁢ with ⁢ ξ > 0 .

Let n∈ℕ and let Yn⊆W1,p⁢(Ω) be an n-dimensional subspace. Then, all norms are equivalent on Yn. So, we can find ρn>0 such that u∈Yn and ∥u∥≤ρn imply

(3.26) | u ⁢ ( z ) | ≤ δ for almost all ⁢ z ∈ Ω .

Using (3.24), (3.25) and (3.26) together with (3.21) and (3.22), for all u∈Yn with ∥u∥≤ρn, we have

(3.27) φ ¯ ⁢ ( u ) ≤ c 9 ⁢ ∥ D ⁢ u ∥ q q + 1 p ⁢ ∫ ∂ ⁡ Ω β ⁢ ( z ) ⁢ | u | p ⁢ 𝑑 σ - ξ q ⁢ ∥ u ∥ q q ≤ ( c 10 - ξ ⁢ c 11 ) ⁢ ∥ u ∥ q

with c10,c11>0 independent of ξ>0 (use the trace theorem and recall that all norms are equivalent on Yn). Recall that ξ>0 is arbitrary (see (2.3)). So, we choose ξ>c10c11 and we have that

φ ⁢ ( u ) < 0 for all ⁢ u ∈ Y n ⁢ with ⁢ ∥ u ∥ = ρ n .

Because of Proposition 3.4 we can apply Theorem 2.1 to find {un}n≥1⊆W1,p⁢(Ω)∖{0} such that

u n ∈ K φ ¯ ∖ { 0 } for all ⁢ n ∈ ℕ

and

φ ¯ ⁢ ( u n ) → 0 as ⁢ n → ∞ .

Since Kφ¯⊆C1⁢(Ω¯) (nonlinear regularity theory), we have

u n ∈ C 1 ⁢ ( Ω ¯ ) ∖ { 0 } for all ⁢ n ∈ ℕ .

Finally, Corollary 3.5 implies that {un}n≥1⊆C1⁢(Ω¯) is a sequence of distinct nodal solutions for (1.1). ∎

Funding statement: V. D. Rădulescu was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-PT-PCCA-2013-4-0614 (PCCA-23/2014).

The authors would like to thank Professor Zhi-Qiang Wang for his interest in this work and for bringing to their attention some relevant related works.

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Received: 2015-08-01

Revised: 2015-12-27

Accepted: 2015-12-28

Published Online: 2016-04-13

Published in Print: 2016-05-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2015-5040

Keywords for this article

Nonhomogeneous Differential Operator; Nodal Solution; Nonlinear Regularity Theory; Nonlinear Maximum Principle

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