Anisotropic problems with unbalanced growth

Ahmed Alsaedi; Bashir Ahmad

doi:10.1515/anona-2020-0063

Article Open Access

Anisotropic problems with unbalanced growth

Ahmed Alsaedi and Bashir Ahmad

Published/Copyright: March 20, 2020

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

The main purpose of this paper is to study a general class of (p, q)-type eigenvalues problems with lack of compactness. The reaction is a convex-concave nonlinearity described by power-type terms. Our main result establishes a complete description of all situations that can occur. We prove the existence of a critical positive value λ^* such that the following properties hold: (i) the problem does not have any entire solution in the case of low perturbations (that is, if 0 < λ < λ^*); (ii) there is at least one solution if λ = λ^*; and (iii) the problem has at least two entire solutions in the case of high perturbations (that is, if λ > λ^*). The proof combines variational methods, analytic tools, and monotonicity arguments.

Keywords: Nonlinear elliptic equation; nonhomogeneous differential operator; variable exponent; weak solution; mountain pass

MSC 2010: 35J60; 35J65; 35J70; 58E05

1 Introduction

The initial motivation of this paper goes back to the paper by Alama and Tarantello [1], where there are studied combined effects of convex and concave terms for nonlinear elliptic equations with Dirichlet boundary condition. Alama and Tarantello were concerned with the following semilinear elliptic problem

−Δu−λu=k(x)uq−h(x)up in Ωu>0 in Ωu=0 on ∂Ω, (1.1)

where 1 < p < q, λ is a real parameter, Ω ⊂ ℝ^N is a smooth bounded domain, and the potentials h, k ∈ L¹(Ω) are nonnegative. Let λ₁ be the first eigenvalue of the Laplace operator in H01 (Ω). The main result in [1] establishes that for all λ ∈ ℝ in a neighbourhood of λ₁, problem (1.1) has nontrivial weak solutions under natural growth hypotheses on h and k. More precisely, Alama and Tarantello proved existence, nonexistence and multiplicity properties depending on λ and according to the integrability properties of the ratio k^p–1/h^q–1.

Nonlinear boundary value problems with reaction described by convex–concave nonlinear terms have been also studied in the pioneering paper by Ambrosetti, Brezis and Cerami [2]. The authors considered the following semilinear elliptic problem with Dirichlet boundary condition

−Δu=λuq−1+up−1 in Ωu>0 in Ωu=0 on ∂Ω, (1.2)

where Ω ⊂ ℝ^N is a bounded domain with smooth boundary, λ is a positive parameter, and 1 < q < 2 < p < 2^* (2^* = 2N/(N – 2) if N ≥ 3, 2^* = +∞ if N = 1, 2). The authors proved the existence of a critical value λ₀ > 0 such that problem (1.2) has at least two solutions for all λ ∈ (0, λ₀), one solution for λ = λ₀, and no solution exists for all λ > λ₀.

For related contributions, we refer to Bartsch and Willem [8], Filippucci, Pucci and Rădulescu [13], Pucci and Rădulescu [23], Rădulescu and Repovš [25], etc.

Motivated by the above mentioned papers, we are concerned with existence and multiplicity properties of solutions in a different abstract setting and with lack of compactness. The feature of the present paper is that we consider a nonlinear Dirichlet problem driven by a general nonhomogenous differential operator, which was introduced by Barile and Figueiredo [5]. This operator generalizes several standard operators, including the p-Laplacian, the (p, q)-Laplace operator, the generalized mean curvature operator, etc. In particular, the associated energy can be a double phase functional.

The study of non-autonomous functionals characterized by the fact that the energy density changes its ellipticity and growth properties according to the point has been initiated by Marcellini [16, 17, 18]. Recently, important contributions are due to Mingione et al. [6, 12]. These papers are in relationship with the works of Zhikov [29, 30], which describe the behavior of phenomena arising in nonlinear elasticity. In fact, Zhikov intended to provide models for strongly anisotropic materials in the context of homogenisation. In particular, he considered the following model functional

Pp,q(u):=∫Ω(|Du|p+a(x)|Du|q)dx,0≤a(x)≤L,1<p<q, (1.3)

where the modulating coefficient a(x) dictates the geometry of the composite made of two differential materials, with hardening exponents p and q, respectively.

Another significant model example of a functional with (p, q)–growth studied by Mingione et al. is given by

u↦∫Ω|Du|plog⁡(1+|Du|)dx,p≥1,

which is a logarithmic perturbation of the p-Dirichlet energy.

Intensive research work has been devoted to nonlinear PDEs with double phase energy; see, e.g., [4, 7, 11, 19, 20, 24, 26, 28].

Some of the main abstract methods used in this paper can be found in Ambrosetti and Rabinowitz [3], Brezis and Nirenberg [10], Pucci and Rădulescu [22], and the monograph by Papageorgiou, Rădulescu and Repovš [21].

Notation

Throughout this paper, we denote by W^1,p(ℝ^N) the Sobolev space endowed with the norm

∥u∥W1,p=∫RN(|∇u|p+|u|p)dx1/p.

We denote by Lms (ℝ^N), 1 ≤ s < ∞ the weighted Lebesgue space

Lms(RN)=u∈Lloc1(RN):∫RNm(x)|u|sdx<∞,

where m(x) is a positive continuous function on ℝ^N. This weighted function space is endowed with the norm

∥u∥m,s=∫RNm(x)|u|sdx1/s.

If m(x) ≡ 1 in ℝ^N, the norm is denoted by ∥⋅∥_s.

2 The main result

Barile and Figueiredo [5] studied nonlinear elliptic problems driven by the potential a : [0, ∞) → (0, ∞), which is a continuously differentiable function satisfying the following hypotheses:

(a1) there exist positive constants c_i(i = 0, 1, 2, 3) and real numbers 1 < p ≤ q such that

c0+c1t(q−p)/p≤a(t)≤c2+c3t(q−p)/pfor allt≥0;

(a2) there exists α ≥ q/p such that

1αa(t)t≤A(t)=∫0ta(s)dsfor allt≥0;

(a3) the mapping [0, ∞) ∋ t ↦ a(t)t^(p–2)/p is increasing.

In this paper, we are concerned with the study of the following nonlinear eigenvalue problem.

−div(a(|∇u|p)|∇u|p−2∇u)+a(|u|p)|u|p−2u=λ|u|r−2u−m(x)|u|s−2u,x∈RNu≥0,x∈RNu≢0, (2.4)

where λ is a positive parameter and

1<p≤q<r<s<p∗. (2.5)

We assume that m : ℝ^N → (0, ∞) is a continuous function satisfying the following (normalized) growth condition

∫RNm(x)r/(r−s)dx=1. (2.6)

As usual, we have denoted by p^* the critical Sobolev exponent, that is, p^* = Np/(N – p) if p < N and p^* = +∞ if p ≥ N.

Let E ⊂ W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) be the function space defined by

E=u∈W1,p(RN)∩W1,q(RN):∫RNm(x)|u|sdx<∞.

Then E is a Banach space endowed with the norm

∥u∥E=∥u∥W1,p+∥u∥W1,q+∥u∥m,s.

We point out that by hypothesis (2.5) and Sobolev embeddings, the space E is continuously embedded into L^r(ℝ^N).

We say that u ∈ E ∖ {0} is a solution of problem (2.4) if u(x) ≥ 0 a.e. in ℝ^N and

∫RNa(|∇u|p)|∇u|p−2∇u∇vdx+∫RNa(|u|p)|u|p−2uvdx= λ∫RN|u|r−2uvdx−∫RNm(x)|u|s−2uvdx, (2.7)

for all v ∈ E.

We state in what follows the main result of this paper. Roughly speaking, this results establishes that problem (2.4) does not have any solutions in the case of small perturbations. However, this problem admits at least two solutions in the case of high perturbations. In both cases, the term “perturbation” should be understood in relationship with the values of the positive parameter λ that is associated to the power-type reaction |u|^r–2u in problem (2.4).

Theorem 1

Assume that hypotheses (2.5) and (a1)–(a3) are fulfilled. Then there exists λ^* > 0 such that the following properties hold.

If 0 < λ < λ^*, then problem (2.4) does not have any solution.
If λ = λ^*, then problem (2.4) has at least one solution.
If λ > λ^*, then problem (2.4) has at least two solutions.

According to Barile and Figueiredo [5], the following operators are suitable to the hypotheses of Theorem 1.

If a ≡ 1, then div(a(|∇u|)|∇u|^p–2∇u) = Δ_pu.
If a(t) = 1 + t^(q–p)/p, then div(a(|∇u|)|∇u|^p–2∇u) = Δ_pu + Δ_qu.
If a(t) = 1 + (1 + t)^–p/(p–2), then

div(a(|∇u|)|∇u|p−2∇u)=Δpu+div|∇u|p−2∇u(1+|∇u|p)(p−2)/p.
If a(t) = 1 + t^(q–p)/p + (1 + t)^–p/(p–2), then

div(a(|∇u|)|∇u|p−2∇u)=Δpu+Δqu+div|∇u|p−2∇u(1+|∇u|p)(p−2)/p.

The energy functional associated to problem (2.4) is 𝓙_λ : E → ℝ defined by

Jλ(u)=1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx−λr∫RN|u|rdx+1s∫RNm(x)|u|sdx.

Recall that A(t) = ∫0t a(s)ds.

Next, by hypothesis (a1), we have

a(t)t(p−1)/p≤c2t(p−1)/p+c3t(q−1)/pfor allt≥0.

This subcritical growth condition implies that 𝓙_λ is well defined. Moreover, by standard arguments, the functional 𝓙_λ is of class C¹ and its Gâteaux directional derivative is given by

〈Jλ′(u),v〉=∫RNa(|∇u|p)|∇u|p−2∇u∇vdx+∫RNa(|u|p)|u|p−2uvdx−λ∫RN|u|r−2uvdx+∫RNm(x)|u|s−2uvdxfor allu,v∈E.

This shows that the nonnegative nontrivial critical points of 𝓙_λ correspond to the solutions of problem (2.4).

Let us assume that u is a solution of problem (2.4). Then the corresponding λ ∈ ℝ is an “eigenvalue” associated to the “eigenfunction” u. This terminology is in accordance with the related notions introduced by Fučik, Nečas, Souček and Souček [14, p. 117] in the context of nonlinear operators. Indeed, if we define the nonlinear operators

Su=1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx−1s∫RNm(x)|u|sdx

and

Tu=1r∫RN|u|rdx

then λ is an eigenvalue for the pair (S, T) (in the sense of [14]) if and only if there is a corresponding eigenfunction u which is a solution of problem (2.4) as described in (2.7).

The strategy to prove Theorem 1 is the following.

We first establish that there is λ^* > 0 such that problem (2.4) has no solution for all λ < λ^* (case of “low perturbations”). This implies that solutions could exist only in the case of “high perturbations”, namely if λ is large enough. The proof of this assertion yields an energy lower bound of solutions in term of λ, which is useful to conclude that problem (2.4) has a non-trivial solution if λ = λ^*.
Next, we show that there exists λ^** > 0 such that problem (2.4) has at least two solutions for all λ > λ^**. Finally, combining the properties of λ^* and λ^** we conclude that λ^* = λ^**.

The proof of Theorem 1 uses some ideas developed in Alama and Tarantello [1], Filippucci, Pucci and Rădulescu [13], and Pucci and Rădulescu [23].

3 Preliminary results

We start with a basic property of the energy functional 𝓙_λ.

Lemma 1

The functional 𝓙_λ is coercive.

Proof

We first observe that using hypotheses (a1) and (a2) we have for all u ∈ E

A(|∇u|p)≥1αa(|∇u|p)|∇u|p≥c0|∇u|p+c1|∇u|qα

and

A(|u|p)≥1αa(|u|p)|u|p≥c0|u|p+c1|u|qα.

Therefore

1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx≥c0α∥u∥W1,pp+c1α∥u∥W1,qq. (3.8)

Next, for fixed a, b ∈ ℝ and 0 < c < d, we consider the mapping

[0,+∞)∋t↦φ(t):=atc−btd.

By straightforward computation we deduce that

φ(t)≤C1aabc/(c−d)for allt≥0,

where C₁ depends only on c, d.

Applying this inequality for a = λ/r, b = m(x)/s, c = r, d = s, we find

λr|u(x)|r−1sm(x)|u(x)|s≤C2λs/(s−r)(m(x))r/(r−s)for allx∈Ω.

By integration and using the normalized assumption (2.6) we obtain

λr∥u∥rr−1s∥u∥m,ss≤C3(λ). (3.9)

Combining relations (3.8) and (3.9), we conclude that 𝓙_λ is coercive.□

Next, with the same arguments as in Barile and Figueiredo [5, p. 460] and the proof of Lemma 2 in Pucci and Rădulescu [23], we can establish that the energy functional 𝓙_λ : E → ℝ is weakly lower semicontinuous.

3.1 Case of low perturbations

In this subsection we prove that solutions of problem (2.4) cannot exist if the positive parameter λ is small enough. This corresponds to the case of “low perturbations”.

Assume that u ∈ E is an eigenfunction of problem (2.4) corresponding to the eigenvalue λ > 0. Choosing v = u in relation (2.7) we obtain

∫RNa(|∇u|p)|∇u|pdx+∫RNa(|u|p)|u|pdx+∫RNm(x)|u|sdx=λ∫RN|u|rdx,

hence

∫RNa(|∇u|p)|∇u|pdx+∫RNa(|u|p)|u|pdx+∥u∥m,ss=λ∥u∥rr. (3.10)

Next, by the Young inequality and using hypothesis (2.5),

λ|u|r≤rsm(x)r/s|u|rs/r+s−rsλm(x)r/ss/(s−r).

Integrating this inequality over Ω we obtain

λ∥u∥rr≤rs∥u∥m,ss+s−rsλs/(s−r)∫RNm(x)r/(r−s)dx.

Thus, by the normalized growth condition (2.6),

λ∥u∥rr≤rs∥u∥m,ss+s−rsλs/(s−r). (3.11)

Combining relations (3.10) and (3.11) we find

∫RNa(|∇u|p)|∇u|pdx+∫RNa(|u|p)|u|pdx≤r−ss∥u∥m,ss+s−rsλs/(s−r).

Using now hypothesis (2.5) we obtain

∫RNa(|∇u|p)|∇u|pdx+∫RNa(|u|p)|u|pdx≤s−rsλs/(s−r). (3.12)

Next, by (a1),

c0tp+c1tq≤a(tp)tpfor allt≥0.

Applying this inequality in relation (3.12) it follows that

c0∫RN|∇u|pdx+c1∫RN|∇u|qdx+c0∫RN|u|pdx+c1∫RN|u|qdx≤s−rsλs/(s−r).

Therefore

c0∥u∥W1,pp+c1∥u∥W1,qq≤s−rsλs/(s−r).

By hypothesis (2.5) we have p < r < p^* and q < r < p^* < q^*. Thus, by the Sobolev embedding theorem, the spaces W^1,p(ℝ^N) and W^1,q(ℝ^N) are continuously embedded into L^r(ℝ^N). It follows that there exists a positive constant C₀ such that

∥u∥rp≤C0∥u∥W1,ppfor allu∈W1,p(RN) (3.13)

and

∥u∥rq≤C0∥u∥W1,qqfor allu∈W1,q(RN). (3.14)

Assuming that u is a solution of problem (2.4), relation (3.10) yields

∫RNa(|∇u|p)|∇u|pdx+∫RNa(|u|p)|u|pdx≤λ∥u∥rr. (3.15)

Using now hypothesis (a1) and relation (3.15), we obtain

c0∥u∥W1,pp+c1∥u∥W1,qq≤λ∥u∥rr.

Thus, by (3.13) and (3.14), there exists a positive constant C₄ not depending on the solution u such that

∥u∥rp+∥u∥rq≤C4∫RNa(|∇u|p)|∇u|pdx+∫RNa(|u|p)|u|pdx≤C4λ∥u∥rr.

This relation implies that

∥u∥r≥max{(C4λ)1/(p−r),(C4λ)1/(q−r)}.

Using this estimate in conjunction with hypothesis (a1) and relations (3.12), (3.13) and (3.14), we obtain that there exists Λ > 0 such that λ > Λ. In particular, solutions do not exist if λ ≤ Λ.

We set

λ∗:=sup{Λ>0:problem (2.4) does not have a solution}. (3.16)

The definition of λ^* implies that problem (2.4) does not have any solution for all λ ∈ (0, λ^*).

4 Proof of Theorem 1

We first prove the existence of two solutions, provided that λ > 0 is large enough. The first solution is obtained by the direct method of the calculus of variations and the corresponding energy is negative. The second solution of problem (2.4) is obtained by applying the mountain pass theorem without the Palais-Smale condition. The energy of this solution is positive.

Since 𝓙_λ is coercive and lower semicontinuous, then it has a global minimizer u₀ ∈ E, see Lemmas 1, 2 and Theorem 1.2 in Struwe [27]. It follows that u₀ is a critical point of 𝓙_λ. In order to show that u₀ is a solution of problem (2.4) it remains to prove that u₀ ≠ 0 and u₀ is nonnegative, provided that λ is sufficiently large.

We first establish that the solution u₀ is nontrivial. For this purpose we show that 𝓙_λ(u₀) < 0.

Consider the following constrained minimization problem

m0:=infu∈Erp∫RN[A(|∇u|p)dx+A(|u|p)]dx+rs∫RNm(x)|u|sdx;∫RN|u|rdx=1.

We observe that m₀ > 0. Indeed, by Hölder’s inequality and hypothesis (2.6), for all u ∈ E with ∥u∥_r = 1

1=∫RN|u|rdx≤∫RNm(x)r/(r−s)(s−r)/r∫RNm(x)|u|sdxr/s=∫RNm(x)|u|sdxr/s,

hence m₀ ≥ r/s > 0.

Fix λ > m₀. Then there exists v ∈ E with ∫RN|v|rdx=1 such that

λ>rp∫RN[A(|∇v|p)+A(|v|p)]dx+rs∫RNm(x)|v|sdx.

Therefore

Jλ(v)=1p∫RNA(|∇v|p)dx+1p∫RNA(|v|p)dx−λr∫RN|v|rdx+1s∫RNm(x)|v|sdx<0.

This shows that inf_u∈E 𝓙_λ(u) < 0 for λ large enough, say for λ > λ_*. In this case, problem (2.4) admits a nontrivial solution u₁, which is a global minimizer of 𝓙_λ. Moreover, since 𝓙_λ (|u₀|) ≤ 𝓙_λ(u₀), we can also assume that u₀ ≥ 0 in Ω.

Our next purpose is to establish the existence of a second solution u₁ ≥ 0 of problem (2.4) for all λ > λ_*. This will be done by using the mountain pass theorem without the Palais-Smale condition of Brezis and Nirenberg [10].

Fix λ > λ_*. Since we are looking for nonnegative solutions, it is natural to consider the truncation

h(x,t)=0ift<0λtr−1−m(x)ts−1if0≤t≤u0(x)λu0r−1−m(x)u0s−1ift>u0(x).

Set H(x, t) = ∫0t h(x, s)ds and consider the C¹-functional 𝓗 : E → ℝ defined by

H(u)=1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx−∫RNH(x,u)dx.

A simple argument shows that 𝓗 is coercive.

The following auxiliary result establishes an interesting location property of the critical points of 𝓗 with respect to the solution u₀.

Lemma 2

If u is an arbitrary critical point of 𝓗, then u ≤ u₀.

Proof

Fix u ∈ E an arbitrary critical point of 𝓗, hence 𝓗′(u) = 0. Since u₀ solves problem (2.4), then Jλ′ (u₀) = 0. We have

0=〈H′(u)−Jλ′(u0),(u−u0)+〉=∫RN(a(|∇u|p)|∇u|p−2∇u−a(|∇u0|p)|∇u0|p−2∇u0)∇(u−u0)+dx+∫RN(a(|u|p)|u|p−2u−a(|u0|p)|u0|p−2u0)(u−u0)+dx−∫RN(h(x,u)−λu0r−1+m(x)u0s−1)(u−u0)+dx.

Taking into account the definition of h, the last integral in the above expression vanishes. It follows that

0=〈H′(u)−Jλ′(u0),(u−u0)+〉=∫[u>u0](a(|∇u|p)|∇u|p−2∇u−a(|∇u0|p)|∇u0|p−2∇u0)∇(u−u0)+dx+∫[u>u0](a(|u|p)|u|p−2u−a(|u0|p)|u0|p−2u0)(u−u0)+dx=∫[u>u0](a(|∇u|p)|∇u|p−2∇u−a(|∇u0|p)|∇u0|p−2∇u0)∇(u−u0)dx+∫[u>u0](a(|u|p)|u|p−2u−a(|u0|p)|u0|p−2u0)(u−u0)dx. (4.17)

Since a(t) ≥ c₀ for all t ≥ 0 (by hypothesis (a1)) and using the monotonicity assumption (a3), we deduce that there exists C₅ > 0 such that for all x, y ∈ ℝⁿ and all n ≥ 1

(a(|x|p)|x|p−2x−a(|y|p)|y|p−2y)⋅(x−y)≥C5|x−y|p.

Combining this inequality with relation (4.17) we obtain

0=〈H′(u)−Jλ′(u0),(u−u0)+〉≥C5∫[u>u0](|∇(u−u0)|p+|u−u0|p)dx≥0.

We conclude that u ≤ u₀.□

We prove in what follows that 𝓗 satisfies the geometric hypotheses of the mountain pass theorem. The existence of a “valley” is guaranteed by the fact that 𝓗 (u₀) = 𝓙_λ(u₀) < 0. The following result establishes the existence of a “mountain” between the origin and u₀.

Lemma 3

There exist positive numbers r and a with r < ∥u₀∥ such that 𝓗(u) ≥ a for all u ∈ E satisfying ∥u∥ = r.

Proof

We have

H(u)=1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx−∫[u>u0]H(x,u)dx−∫[0≤u≤u0]H(x,u)dx=1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx−λr∫[u>u0]u0rdx−λr∫[0≤u≤u0]urdx≥1p∫RNA(|∇u|p)dx+1p∫RNA(|u|p)dx−λr∫RNu0rdx (4.18)

On the one hand, by hypotheses (a1) and (a2) we find

A(|∇u|p)≥1αa(|∇u|p)|∇u|p≥1α(c0|∇u|p+c1|∇u|q). (4.19)

and

A(|u|p)≥1αa(|u|p)|u|p≥1α(c0|u|p+c1|u|q). (4.20)

On the other hand, by the Sobolev embedding theorem, there exists C₆ > 0 such that

∥u∥r≤C6∥u∥W1,pfor allu∈E. (4.21)

Combining relations (4.18), (4.19), (4.20), (4.21) and hypothesis (2.5), we deduce that there exist r < ∥u₀∥ small enough and a > 0 such that 𝓗(u) ≥ a for all u ∈ E satisfying ∥u∥ = r.□

Set

P={p∈C([0,1],E);p(0)=0andp(1)=u0}

and

c=infp∈Pmaxt∈[0,1]H(p(t))>0.

Applying the mountain pass theorem without the Palais-Smale condition, there exists a sequence (z_n) in E such that

H(zn)→casn→∞ (4.22)

and

∥H′(zn)∥E∗→0asn→∞. (4.23)

By relation (4.22) and since 𝓗 is coercive, we deduce that (z_n) is bounded. So, there exists u₁ ∈ E such that, up to a subsequence,

zn⇀u1inE.

Using this information in conjunction with (4.23) we deduce that

H′(zn)(φ)→H′(u1)(φ)for allφ∈Cc∞(RN).

Since Cc∞ (ℝ^N) is dense in W^1,p(ℝ^N) ∩ W^1,q(ℝ^N) and E is continuously embedded into W^1,p(ℝ^N) ∩ W^1,q(ℝ^N), we deduce that

H′(zn)(v)→H′(u1)(v)for allv∈E.

By (4.23) we obtain that 𝓗′(u₁) = 0, hence u₁ is a solution of problem (2.4). We conclude that problem (2.4) admits at least two solutions for all λ > λ_*.

Set

λ∗∗:=inf{λ>0;problem (2.4) has a solution}.

Recall that

λ∗:=sup{λ>0;problem (2.4) does not have any solution}.

Then 0 < λ^* ≤ λ^** < ∞.

To complete the proof, we need to argue the following assertions:

problem (2.4) has at least two solutions for all λ > λ^**;
λ^* = λ^** and problem (2.4) has a solution for λ = λ^*.

Assertion (i) follows by standard arguments based on the monotonicity hypothesis (a3). Assertion (ii) is a consequence of the fact that problem (2.4) does not have any solution provided that λ < λ^**. In both cases we refer for details to the proof of Theorem 1.1 in [13].

If we replace hypothesis (2.5) with

1<p≤q<s<r<p∗,

then the associated energy functional is no longer coercive but has a mountain pass geometry for all λ > 0. A straightforward argument shows the following properties:

problem (2.4) does not have any solution if λ ≤ 0;
problem (2.4) has at least one solution for all λ > 0.

In this case we can apply the mountain pass theorem of Ambrosetti and Rabinowitz. The mountain pass geometry of the problem is generated by the assumption 1 < p ≤ q < s < r < p^*. We also point out that since s < r, then any Palais-Smale sequence is bounded in E. The details of the proof are left to the reader.

Acknowledgments

This project was funded by the Research and Development Office (RDO) at the Ministry of Education, Kingdom of Saudi Arabia under the Grant No. (HIQI-3-2019). The authors also acknowledge with thanks the Research and Development Office at King Abdulaziz University (RDO-KAU) for technical support.

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Received: 2019-10-16

Accepted: 2019-11-08

Published Online: 2020-03-20

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

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0063

Keywords for this article

Nonlinear elliptic equation; nonhomogeneous differential operator; variable exponent; weak solution; mountain pass

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