The existence of solution for a fourth-order nonlinear partial differential equation (PDE) class involving p-biharmonic operator Ξ”(|Δ𝑒|π‘βˆ’2Δ𝑒)=πœ†πœŒ(π‘₯)|𝑒|π‘žβˆ’2𝑒inΞ©,𝑒=Δ𝑒=0,onπœ•Ξ©, is proved by applying mountain pass theorem and a local minimization.

1. Introduction and Assumptions

The goal of this paper is to investigate the existence of nontrivial solutions for a class of nonlinear partial differential equations of the fourth order of the form Ξ”ξ‚€||||Ξ”π‘’π‘βˆ’2Δ𝑒=πœ†πœŒ|𝑒|π‘žβˆ’2𝑒inΞ©,𝑒=Δ𝑒=0onπœ•Ξ©,(𝐸) where Ξ©βŠ‚β„π‘ is a smooth bounded domain in ℝ𝑁, πœ†βˆˆβ„+ is a parameter which plays the role of eigenvalue, and 𝑝 and π‘ž are real numbers such that 𝑝>1 and 1<π‘ž<π‘βˆ—βˆ—if𝑁1<𝑝<2,π‘ž<+∞if𝑁𝑝β‰₯2,(1.1) where π‘βˆ—βˆ— is the critical Sobolev exponent defined by π‘βˆ—βˆ—=𝑁𝑝/(π‘βˆ’2𝑝) if 1<𝑝<𝑁/2 and +∞ if 𝑝β‰₯𝑁/2, and 𝜌 is an indefinite weight in πΏπ‘Ÿ(Ξ©) with π‘Ÿ being so that (i)if𝑁1<𝑝<2,then⎧βŽͺ⎨βŽͺβŽ©π‘π‘Ÿ>2π‘žfor1<π‘žβ‰€π‘,π‘Ÿ>π‘βˆ—βˆ—ξ€·π‘βˆ—βˆ—ξ€Έβˆ’π‘žβˆ’1for𝑝<π‘ž<π‘βˆ—βˆ—,(ii)π‘Ÿ>π‘žif𝑁𝑝=2,(iii)π‘Ÿ=1if𝑁𝑝>2,(1.2) and (the Lebesgue measure) mes({π‘₯∈Ω∢𝜌(π‘₯)>0})β‰ 0.(1.3)Ξ”2π‘π‘’βˆΆ=Ξ”(|Δ𝑒|π‘βˆ’2Δ𝑒) is an operator of the fourth order called the 𝑝-biharmonic operator. For 𝑝=2, the linear operator Ξ”2=ΔΔ is the iterated Laplace that multiplied with positive constant occurs often in Navier-Stokes equations as being a viscosity coefficient. Its reciprocal operator denoted (Ξ”2)βˆ’1 is the celebrated Green operator [1]. The boundary condition in 𝐸 is of the compatibility type when 𝑝=π‘ž which can be considered as the following Hamiltonian system of the two coupled equations: βˆ’Ξ”π‘’=πœ™π‘(𝑣)inΞ©,βˆ’Ξ”π‘£=πœ†πœŒπœ™π‘(𝑒)inΞ©,𝑒=𝑣=0(1.4) with πœ™π‘(𝑑)=|𝑑|π‘βˆ’2𝑑, and π‘ξ…ž=𝑝/(π‘βˆ’1) is the HΓΆlder conjugate exponent of 𝑝, by a transformation of a problem to a known Poisson's problem and using the properties of the operator-solution stated by Agmon-Douglis-Nirenberg; see [2]. Notice that a similar system as was considered recently by [3] in the restrictive case πœŒβ‰‘1. Our approach it quite different. Note that, in the case 𝑝=π‘ž, 𝐸 is (π‘βˆ’1)-homogeneous, so we are dealing a quasilinear eigenvalue problem which is considered differently. This is few considered by [4] in the particular case πœŒβ‰‘1 and Ξ© is smooth; and by [5, 6] with π‘’βˆˆπ‘Š02,𝑝(Ξ©) as a boundary condition, for any bounded domain.  

Note also that the nonhomogeneous case is not considered there. Here we seek nontrivial solutions for 𝐸 by distinguishing between two subcritical cases 1<π‘ž<𝑝 and 𝑝<π‘ž<π‘βˆ—βˆ— which means that the critical points of the associated energy functional1π’œ(𝑒)=π‘ξ€œΞ©||||Ξ”π‘’π‘πœ†π‘‘π‘₯βˆ’π‘žξ€œΞ©πœŒ(π‘₯)|𝑒|π‘žπ‘‘π‘₯,(1.5)are defined in 𝑋=π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©). One solution is obtained by applying classical Mountain Pass Theorem and the other solution by a local minimization technique. The restrictive case πœŒβ‰‘1 and Ξ”2𝑝 is substituted by the well-known 𝑝-Laplacian βˆ’Ξ”π‘ was studied by Azorero and Alonso [7] and Elkhalil and Touzani [8] by using the fundamental properties of the first eigenvalue of the Dirichlet 𝑝-Laplacian problem.

It is important to indicate here that condition (1.2) is optimal to ensure the Palais-Smale (PS) condition is satisfied. Notice that our results are investigated without any condition on the parameter πœ† related to the spectrum of 𝐸 (when 𝑝=π‘ž). On the other hand the condition (1.3) is assumed to have eventually nontrivial solutions.

The rest of this paper is divided in two sections as follows. In Section 2, we introduce some preliminary results and we give some technical lemmas, and in Section 3, we establish our main results.

2. Preliminaries

First, we introduce some preliminary results that we will need and some lemmas that are the key point of our results. We solve the problem 𝐸 in the space 𝑋=π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) endowed with the norm β€–ξ‚΅ξ€œπ‘’β€–=Ξ©||||Δ𝑒𝑝𝑑π‘₯1/𝑝.(2.1)𝑋 equipped with this norm is an uniformly convex Banach space for 1<𝑝<∞. Hereafter, ‖⋅‖𝑝 is the 𝐿𝑝-norm, βŸ¨β‹…,β‹…βŸ© will denote the duality between 𝑋 and its dual π‘‹ξ…ž.

Definition 2.1. 𝑒 is a weak solution of 𝐸 if, and only if, for all π‘£βˆˆπ‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) we have ξ€œΞ©||||Ξ”π‘’π‘βˆ’2ξ€œΞ”π‘’Ξ”π‘£π‘‘π‘₯=πœ†Ξ©πœŒ|𝑒|π‘žβˆ’2𝑒𝑣𝑑π‘₯.(2.2)

That is, 𝑒 is a critical point of the energy functional associated to 𝐸 defined on 𝑋 by 1π’œ(𝑒)=π‘ξ€œΞ©||||Ξ”π‘’π‘πœ†π‘‘π‘₯βˆ’π‘žξ€œΞ©πœŒ|𝑒|π‘žπ‘‘π‘₯.(2.3)

Lemma 2.2. For any π‘Ÿ verifying (1.2) and π‘ž satisfying (1.1) there exists a constant 𝑐=𝑐(𝑝,π‘ž,π‘Ÿ)>0 such that ||||ξ€œΞ©πœŒ|𝑒|π‘ž||||𝑑π‘₯β‰€π‘β€–πœŒβ€–π‘Ÿβ€–π‘’β€–π‘ž(2.4) and the map π‘’β†’πœŒ|𝑒|π‘βˆ’2𝑒 is strongly continuous from 𝑋 into π‘‹ξ…ž.

Proof. To establish (2.4), we will divide the proof to three steps with respect to exponents 𝑝,π‘ž and 𝑁.Step 1 (1<𝑝<𝑁). Fixing π‘’βˆˆπ‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) and using HΓΆlder's inequality we have ||||ξ€œπœŒ|𝑒|π‘ž||||𝑑π‘₯β‰€β€–πœŒβ€–π‘Ÿβ€–π‘’β€–π‘ π‘žβˆ’1β€–π‘’β€–π‘βˆ—βˆ—,(2.5) where π‘žβˆ’1𝑠1=1βˆ’π‘Ÿβˆ’1π‘βˆ—βˆ—.(2.6) Such exponent 𝑠 exists. Indeed: if 1<π‘žβ‰€π‘, we obtain π‘βˆ’1𝑠β‰₯π‘žβˆ’1𝑠1=1βˆ’π‘Ÿβˆ’1π‘βˆ—βˆ—β‰₯1βˆ’2π‘žπ‘βˆ’1π‘βˆ—βˆ—β‰₯1βˆ’2π‘π‘βˆ’1π‘βˆ—βˆ—=π‘βˆ’1π‘βˆ—βˆ—.(2.7) Thus, it suffices to take 𝑠 so that max(1,π‘žβˆ’1)<𝑠<π‘βˆ—βˆ—.(2.8) If 𝑝<π‘ž<π‘βˆ—βˆ—, we have π‘Ÿ>π‘βˆ—βˆ—(π‘βˆ—βˆ—βˆ’π‘ž)βˆ’1. So, 1π‘Ÿπ‘ž<1βˆ’π‘βˆ—βˆ—.(2.9) Hence, π‘žβˆ’1𝑠1=1βˆ’π‘Ÿβˆ’1π‘βˆ—βˆ—>π‘žβˆ’1π‘βˆ—βˆ—.(2.10) Therefore, it suffices to take 𝑠 such that max(1,π‘žβˆ’1)<𝑠<π‘βˆ—βˆ—.(2.11)Step 2 (𝑝=𝑁/2). In this case, for any 𝑠>1, π‘Š01,π‘βˆ©π‘Š2,𝑝(Ξ©)β†ͺ𝐿𝑠(Ξ©). Thus, for 𝑑=1/(1βˆ’(π‘Ÿ+π‘žξ…ž)/π‘Ÿπ‘žξ…ž) (some 𝑑 exists because π‘Ÿ>π‘ž=π‘žξ…ž/(π‘žξ…žβˆ’1)), we get 1𝑑+1π‘žξ…ž+1π‘Ÿ=1.(2.12) HΓΆlder's inequality yields ||||ξ€œπœŒ|𝑒|π‘ž||||β‰€β€–πœŒβ€–π‘Ÿβ€–π‘’β€–π‘‘β€–π‘’β€–π‘žβ€².(2.13)Step 3 (𝑝>𝑁/2). In this case we have π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©)β†ͺ𝐿∞(Ξ©) and π‘Ÿ=1. Thus, ξ€œΞ©πœŒ|𝑒|π‘žπ‘‘π‘₯β‰€β€–π‘’β€–π‘žβˆžβ€–πœŒβ€–1β‰€π‘β€–π‘’β€–π‘žβ€–πœŒβ€–1.(2.14) To prove the continuity of π‘’β†’πœŒ|𝑒|π‘žβˆ’2𝑒, we argue as follows.
Let (𝑒𝑛)𝑛β‰₯0βŠ‚π‘‹ such that 𝑒𝑛⇀𝑒 in 𝑋. Thus, 𝜌|𝑒𝑛|π‘žβˆ’2π‘’π‘›β†’πœŒ|𝑒|π‘žβˆ’2𝑒 in π‘‹ξ…ž, that is, supπ‘£βˆˆπ‘‹,‖𝑒‖≀1||||ξ€œΞ©πœŒξ‚ƒ||𝑒𝑛||π‘žβˆ’2π‘’π‘›βˆ’|𝑒|π‘žβˆ’2𝑒||||𝑣𝑑π‘₯=π‘œ(1).(2.15) We prove it in the case 1<𝑝<𝑁/2, 1<π‘ž<π‘βˆ—βˆ— and πœŒβˆˆπΏπ‘Ÿ(Ξ©) with π‘Ÿ satisfying (i).
Using (2.6) or (2.10), we obtain by calculation supπ‘£βˆˆπ‘‹,‖𝑣‖1,𝑝≀1||||ξ€œΞ©πœŒξ‚ƒ||𝑒𝑛||π‘žβˆ’2π‘’π‘›βˆ’|𝑒|π‘žβˆ’2𝑒||||𝑣𝑑π‘₯≀supπ‘£βˆˆπ‘‹,‖𝑣‖1,𝑝≀1ξ‚Έβ€–πœŒβ€–π‘Ÿβ€–β€–ξ‚€||𝑒𝑛||π‘žβˆ’2π‘’π‘›βˆ’|𝑒|π‘žβˆ’2𝑒‖‖𝑠/(π‘žβˆ’1)β€–π‘£β€–π‘βˆ—βˆ—ξ‚Ήβ‰€π‘β€–πœŒβ€–π‘Ÿβ€–β€–ξ‚€||𝑒𝑛||π‘žβˆ’2π‘’π‘›βˆ’|𝑒|π‘žβˆ’2𝑒‖‖𝑠/(π‘žβˆ’1).(2.16) Therefore, the desired result can be obtained since the limit β€–β€–ξ‚€||𝑒𝑛||π‘žβˆ’2π‘’π‘›βˆ’|𝑒|π‘žβˆ’2𝑒‖‖𝑠/(π‘žβˆ’1)=π‘œ(1)(2.17) holds by using the continuity of the Nemytskii operator 𝑒→|𝑒|π‘žβˆ’2𝑒 from 𝐿𝑠(Ξ©)→𝐿𝑠/(π‘žβˆ’1)(Ξ©) and the right embedding of Sobolev space.

Remark 2.3. If 𝑒 is a solution of 𝐸 associated to πœ†=1, then, for any 𝛼>0, 𝛼1/(π‘βˆ’π‘ž)𝑒 is also a weak solution of 𝐸 associated to 𝛼. Hence, we can reduce the problem to πœ†=1.

Lemma 2.4. For π‘βˆˆ(1,+∞) we have the following assertions (i)if 1<π‘ž<𝑝 then π’œ is bounded from below, (ii)if 𝑝<π‘ž<π‘βˆ—βˆ— thus, there exist two reel 𝜎>0 and 𝛿>0 such that:𝐴(𝑒)>𝛿 if 0<‖𝑒‖<𝜎,𝐴(𝑒)β‰₯0 if ‖𝑒‖=𝜎.

Proof. Step 1 (1<π‘ž<𝑝). We have1π’œ(𝑒)=π‘β€–π‘’β€–π‘βˆ’1π‘žξ€œΞ©πœŒ|𝑒|π‘žβ‰₯1π‘β€–π‘’β€–π‘βˆ’π‘π‘žβ€–πœŒβ€–π‘Ÿβ€–π‘’β€–π‘žβ‰₯β€–π‘’β€–π‘žξ‚Έ1π‘β€–π‘’β€–π‘βˆ’π‘žβˆ’π‘π‘žβ€–πœŒβ€–π‘Ÿξ‚Ή,(2.18) for a positive constant 𝑐.
If ‖𝑒‖β‰₯((𝑐𝑝/π‘ž)β€–πœŒβ€–π‘Ÿ)1/(π‘βˆ’π‘ž), then π’œ(𝑒)β‰₯0.(2.19)
If ‖𝑒‖<((𝑐𝑝/π‘ž)β€–πœŒβ€–π‘Ÿ)1/(π‘βˆ’π‘ž), thenπ’œ(𝑒)β‰₯π‘ξ…žwithπ‘ξ…žπ‘=βˆ’π‘žβ€–πœŒβ€–π‘Ÿξ‚΅π‘π‘π‘žβ€–πœŒβ€–π‘Ÿξ‚Ά1/(π‘βˆ’π‘ž).(2.20)
Step 2 (𝑝<π‘ž<π‘βˆ—βˆ—). From (2.18) we deduce that 1𝐴(𝑒)β‰₯π‘β€–π‘’β€–π‘žξ‚Έ1βˆ’π‘π‘π‘žβ€–πœŒβ€–π‘Ÿβ€–π‘’β€–π‘žβˆ’π‘ξ‚Ή.(2.21) If ‖𝑒‖≀((π‘ž/𝑝𝑐)(1/β€–πœŒβ€–π‘Ÿ))1/(π‘žβˆ’π‘), then there exists 𝛿>0 such that π’œ(𝑒)>𝛿.(2.22) If ‖𝑒‖=((π‘ž/𝑝𝑐)(1/β€–πœŒβ€–π‘Ÿ))1/(π‘žβˆ’π‘), we get π’œ(𝑒)>0.(2.23) The estimation above completes the proof.

Proposition 2.5. If the pair (𝑝,π‘ž) satisfies (1.1) with π‘β‰ π‘ž, then π’œ satisfies the (PS) condition.

Proof. Let (𝑒𝑗)𝑗 be a sequence of Palais-Smale of π’œ in 𝑋. Thus there exists 𝑀>0 such that ||π’œξ€·π‘’π‘—ξ€Έ||(≀𝑀PS)1 and for any πœ€>0 there is 𝑗0>0 such that ||ξ«π’œξ…žξ€·π‘’π‘—ξ€Έ,𝑒𝑗||β€–β€–π‘’β‰€πœ€π‘—β€–β€–,βˆ€π‘—β‰₯𝑗0.(PS)2 Claim that (𝑒𝑗)𝑗 is a bounded sequence in 𝑋.Step 1 (1<𝑝<π‘ž). By (2.18) we deduce that π’œ is coercive, then the claim follows.Step 2 (𝑝<π‘ž<π‘βˆ—βˆ—). Equation (PS)2 implies that, for all 𝑗>𝑗0, β€–β€–π‘’βˆ’πœ€π‘—β€–β€–β‰€ξ«π’œξ…žξ€·π‘’π‘—ξ€Έ,π‘’π‘—ξ¬β€–β€–π‘’β‰€πœ€π‘—β€–β€–.(2.24) Thanks to (PS)1, we obtain, 1βˆ’π‘€β‰€π‘ξ€œΞ©||Δ𝑒𝑗||𝑝1𝑑π‘₯βˆ’π‘žξ€œπœŒ||𝑒𝑗||π‘žπ‘‘π‘₯≀𝑀.(2.25) Multiplying (2.24) by (βˆ’1/π‘ž) and adding with (2.18), we obtain, ξ‚΅1π‘βˆ’1π‘žξ‚Άβ€–β€–π‘’π‘—β€–β€–π‘πœ€β‰€π‘€+π‘žβ€–β€–π‘’π‘—β€–β€–.(2.26) Hence, there exists a positive constant 𝑐>0 such that β€–β€–π‘’π‘—β€–β€–β‰€π‘βˆ€π‘—β‰₯𝑗0.(2.27) So (𝑒𝑗) is bounded in the two cases in 𝑋 and the claim is concluded. Consequently, there exists a subsequence still denoted (𝑒𝑗)𝑗 converges weakly for some π‘’βˆˆπ‘‹ and strongly in 𝐿𝑝(Ξ©) and in πΏπ‘ž(Ξ©) for all (𝑝,π‘ž) satisfies (1.2).
Let ξ€œβŸ¨π½π‘’,π‘£βŸ©=Ξ©||||Ξ”π‘’π‘βˆ’2Δ𝑒Δ𝑣𝑑π‘₯.(2.28) Thus ξ«π΄ξ…žξ€·π‘’π‘—ξ€Έβˆ’π΄ξ…ž(𝑒),𝑒𝑗+ξ€œβˆ’π‘’Ξ©πœŒξ‚€||𝑒𝑗||π‘žβˆ’2π‘’π‘—βˆ’|𝑒|π‘žβˆ’2𝑒𝑒𝑗=ξ«βˆ’π‘’π½π‘’π‘—βˆ’π½π‘’,𝑒𝑗.βˆ’π‘’(2.29) Since π‘’β†’πœŒ|𝑒|π‘žβˆ’2𝑒 is strongly continuous from 𝑋 into π‘‹ξ…ž, we deduce from (PS)2, as 𝑗→+∞ in (2.29), that lim𝑗→+βˆžξ«π½π‘’π‘—βˆ’π½π‘’,π‘’π‘—ξ¬βˆ’π‘’=0.(2.30) On the other hand, we have ξ«π½π‘’π‘—βˆ’π½π‘’,𝑒𝑗β‰₯ξ‚€β€–β€–βˆ’π‘’Ξ”π‘’π‘—β€–β€–π‘π‘βˆ’1βˆ’β€–Ξ”π‘’β€–π‘π‘βˆ’1β€–β€–ξ‚ξ‚€Ξ”π‘’π‘—β€–β€–π‘βˆ’β€–Ξ”π‘’β€–π‘ξ‚.β‰₯0.(2.31) This and (2.30) imply that β€–β€–Ξ”π‘’π‘—β€–β€–π‘βŸΆβ€–Ξ”π‘’β€–π‘asπ‘—βŸΆ+∞.(2.32) Finally, since π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) is uniformly convex, the proof is achieved.

3. Main Results

3.1. The Case 1<π‘ž<𝑝: Local Minimization Technique

In this subsection, we show that the problem 𝐸 has a sequence of weak solution by using the results of Lusternik-Shnirleman [9]. In other words, we use a local minimization for the corresponding energy functional.

Let Γ𝑛={π΅βŠ‚π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©),compactsymmetricand𝛾(𝐡)β‰₯𝑛} and set 𝑐𝑛=infπ΅βˆˆΞ“π‘›ξ‚΅supπ‘’βˆˆπ΅ξ‚Άπ’œ(𝑒),βˆ€π‘›βˆˆβ„•βˆ—.(3.1) We will show that the sequence 𝑐𝑛, defined by (3.1) is critical values of π’œ. Here and in the following 𝛾(𝐡)=π‘˜ is the genus of 𝐡, that is, the smallest integer π‘˜ such that there is an odd continuous map from 𝐡 into β„π‘˜β§΅{0}.

Lemma 3.1. For any π‘˜βˆˆβ„•βˆ—, Ξ“π‘˜β‰ βˆ….(3.2)

Proof. π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) is separable. Therefore, for any π‘˜βˆˆβ„•βˆ—, there exists a finite sequence of functions 𝑒1,…,π‘’π‘˜ in π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) linearly independent such that suppπ‘’π‘–βˆ©supp𝑒𝑗=βˆ… for 𝑖≠𝑗 and thanks to (1.3) we can choose 𝑒𝑖 such that βˆ«Ξ©π‘”|𝑒𝑖|π‘žπ‘‘π‘₯=1. Let πΉπ‘˜=span(𝑒1,…,π‘’π‘˜) be a subspace in π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) spend by 𝑒𝑖 of dimension π‘˜.
If π‘£βˆˆπΉπ‘˜, then, there exist 𝛽1,…,π›½π‘˜ real numbers such that βˆ‘π‘£=𝑖=π‘˜π‘–=1𝛽𝑖𝑒𝑖.
Thus, ∫Ω𝜌|𝑣|π‘žβˆ‘π‘‘π‘₯=𝑖=π‘˜π‘–=1|𝛽𝑖|π‘ž. Hence, the mapξ‚΅ξ€œπ‘£βŸΆΞ©πœŒ|𝑒|π‘žξ‚Άπ‘‘π‘₯1/π‘ž(3.3) is a norm in πΉπ‘˜. Consequently, there exists a constant 𝑐>0 such that ξ‚΅ξ€œπ‘β€–π‘’β€–β‰€πœŒ|𝑒|π‘žξ‚Άπ‘‘π‘₯1/π‘žβ‰€1𝑐‖𝑒‖,βˆ€π‘£βˆˆπΉπ‘˜.(3.4) Set π‘ˆπ‘˜=πΉπ‘˜βˆ©ξ‚»πœƒπ‘£βˆˆπ‘‹βˆΆ2β‰€ξ€œΞ©πœŒ|𝑣|π‘žξ‚Όπ‘‘π‘₯β‰€πœƒ(3.5) with πœƒ=(𝑝/π‘žπ‘π‘)π‘ž/(π‘βˆ’1). It is clear that π‘ˆπ‘˜ is a closed neighborhood, symmetric, compact not containing 0. Finally, by the property of genus we get 𝛾(π‘ˆπ‘˜)=π‘˜ and Ξ“π‘˜β‰ βˆ….

Theorem 3.2. Let 𝑐𝑛 be a sequence defined by (3.1). Then, we have the following statements: (i)For any π‘›βˆˆβ„•βˆ—, there exist 𝑛 distinct pairs critical points of the functional π’œ. (ii)If βˆ’βˆž<𝑐=𝑐𝑛=𝑐𝑛+1=β‹―=𝑐𝑛+π‘˜(3.6) then 𝛾𝐾𝑐β‰₯π‘˜+1,(3.7) where 𝐾𝑐=ξ‚†π‘’βˆˆπ‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©)suchthatπ’œξ…žξ‚‡.(𝑒)=0,π’œ(𝑒)=𝑐(3.8)

Proof. From the result of [10], it suffices to prove (i).
Recall that the functional 𝐴 is even, bounded from below and sup𝑋𝐴(𝑒)<0.
In view of Lemma 2.4, 𝐴 is of class 𝐢1 on 𝑋; satisfying the (PS) condition.
Hence, these properties confirm the hypotheses of Clark's Lemma cf. [11] which proves (i).

Remark 3.3. The set 𝐾={π‘’βˆˆπ‘Š01,𝑝(Ξ©)βˆ©π‘Š1,𝑝(Ξ©)βˆΆπ’œξ…ž(𝑒)=0} is compact. Indeed, let (𝑒𝑛)𝑛 be a sequence in 𝐾, that is, π’œξ…žξ€·π‘’π‘›ξ€Έ=0,βˆ€π‘›βˆˆβ„•.(3.9) It is clear that, for 𝑝>π‘ž, π’œξ€·π‘’π‘›ξ€Έβ‰€ξ«π’œξ…žξ€·π‘’π‘›ξ€Έ,𝑒𝑛=0,βˆ€π‘›.(3.10) Since 𝐴 is bounded from below, by Lemma 2.4, (𝑒𝑛)𝑛 is a sequence of Palais-Smale (PS). Thus, (𝑒𝑛)𝑛 possesses a convergent subsequence.
Consequently, 𝐾 is compact in π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝(Ξ©) and the functional π’œ has an infinity of critical points.

3.2. The Case 𝑝<π‘ž<π‘βˆ—βˆ—: Mountain Pass Theorem

Here, we prove the existence of solutions of problem 𝐸, by using Mountain Pass theorem [11].

Lemma 3.4. Suppose that 𝑝<π‘ž<π‘βˆ—βˆ— and πœŒβˆˆπΏπ‘Ÿ(Ξ©) with π‘Ÿ satisfying (1.2). Hence there exists π‘’βˆˆπ‘‹β§΅{0} such that π’œ(𝑒)≀0.(3.11)

Proof. We have for any π‘’βˆˆπ‘‹ and π‘‘βˆˆβ„, π‘‘π’œ(𝑑𝑒)=π‘π‘ξ€œΞ©||||Δ𝑒𝑝𝑑𝑑π‘₯βˆ’π‘žπ‘žξ€œΞ©πœŒ|𝑒|π‘žπ‘‘π‘₯.(3.12) From (1.3), there exists 𝑒0βˆˆπ‘‹ such that ∫Ω𝜌|𝑒0|π‘žπ‘‘π‘₯=1. Thus, lim𝑑→+βˆžπ’œξ€·π‘‘π‘’0ξ€Έ=βˆ’βˆž,because𝑝<π‘ž.(3.13) Hence, there is 𝑑0>0 large enough so that π’œ(𝑑0𝑒0)<0. To achieve the proof, it suffices to take 𝑒=𝑑0𝑒0.

Consequently, we conclude the following statements. (a)π’œ is 𝐢1 of class 𝐢1, even and π’œ(0)=0. (b)π’œ satisfies the (PS) condition.(c)There exist two positive reel 𝑙>0, 𝛿>0 such that π’œ(𝑒)>𝛿ifπ’œ0<‖𝑒‖<𝑙,(𝑒)β‰₯0if‖𝑒‖=𝑙.(3.14)(d)There exists π‘’βˆˆπ‘‹β§΅{0} such that π’œ(𝑒)≀0.

Now, we can establish the following theorem.

Theorem 3.5. If 𝑝<π‘ž<π‘βˆ—βˆ—, then the value 𝑐 defined by 𝑐=infπ›Ύβˆˆπ’’maxπ‘‘βˆˆ[0,1]π’œ(𝛾(𝑑))(3.15) is a critical value of π’œ.
Here, []𝒒=π›ΎβˆˆπΆ0,1;π‘Š01,𝑝(Ξ©)βˆ©π‘Š2,𝑝.(Ξ©),𝛾(0)=0,𝛾(1)=𝑒(3.16)

Proof. In view of Lemma 2.4., Proposition 2.5, and properties (a)–(d), 𝑐 is a critical value of 𝐴 by applying the Mountain Pass Theorem due to [11].


The author gratefully acknowledges the financial support provided by Al-Imam Muhammed Ibn Saud Islamic University during this research.