Multiplicity of positive solutions for (p,q) -Laplace equations with two parameters
aa r X i v : . [ m a t h . A P ] J u l Multiplicity of positive solutions for ( p, q ) -Laplace equationswith two parameters Vladimir Bobkov
Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West BohemiaUniverzitní 8, 301 00 Plzeň, Czech RepublicInstitute of Mathematics, Ufa Federal Research Centre, RASChernyshevsky str. 112, 450008 Ufa, Russiae-mail: [email protected]
Mieko Tanaka
Department of Mathematics, Tokyo University of ScienceKagurazaka 1-3, Shinjyuku-ku, Tokyo 162-8601, Japane-mail: [email protected]
Abstract
We study the zero Dirichlet problem for the equation − ∆ p u − ∆ q u = α | u | p − u + β | u | q − u in a bounded domain Ω ⊂ R N , with < q < p . We investigate the relationbetween two critical curves on the ( α, β ) -plane corresponding to the threshold of existenceof special classes of positive solutions. In particular, in certain neighbourhoods of the point ( α, β ) = (cid:0) k∇ ϕ p k pp / k ϕ p k pp , k∇ ϕ p k qq / k ϕ p k qq (cid:1) , where ϕ p is the first eigenfunction of the p -Laplacian, we show the existence of two and, which is rather unexpected, three distinctpositive solutions, depending on a relation between the exponents p and q . Keywords : ( p, q ) -Laplacian, positive solutions, fibered functional, mountain pass theo-rem, local minimum, S-shaped bifurcation, three solutions. MSC2010 : 35J62, 35J20, 35P30, 35B50
1. Introduction and main results
We consider the boundary value problem ( − ∆ p u − ∆ q u = α | u | p − u + β | u | q − u in Ω ,u = 0 on ∂ Ω , ( D α,β )where the operator ∆ r , formally defined as ∆ r u = div (cid:0) |∇ u | r − ∇ u (cid:1) for r = p, q > , is the r -Laplacian, α, β ∈ R are parameters, and Ω ⊂ R N is a bounded domain, N ≥ . In the case N ≥ , we require the boundary ∂ Ω of Ω to be C -smooth. Throughout the text, we alwaysassume q < p , which involves no loss of generality.The differential operator in the problem ( D α,β ) is usually called the ( p, q ) -Laplacian, andthereby ( D α,β ) can be formally understood as the corresponding eigenvalue problem. Althoughthe presence of two spectral parameters ( α and β ) is not typical in nonlinear spectral theories(cf. [3, 18]), such choice appears to be more convenient for our particular problem since itprovides a separate control of the influence of the ( p − - and ( q − -homogeneous parts.Considered independently, these parts correspond to the eigenvalue problems for the p - and1 -Laplacians, and it is thus natural to anticipate a strong dependence of the structure of thesolution set of ( D α,β ) on the spectrum of the both p - and q -Laplacians. Indeed, the problem( D α,β ) has been investigated in a few works, where certain nontrivial dependences of this kindwere obtained, see, e.g., [14, 15, 22, 27, 30, 40], the works [5, 6, 7, 8] of the present authors,and a survey [26]. In the present article, we continue our investigation of the problem ( D α,β )by establishing several nontrivial multiplicity results, mainly in certain neighbourhoods of thepoint ( α, β ) = (cid:18) k∇ ϕ p k pp k ϕ p k pp , k∇ ϕ p k qq k ϕ p k qq (cid:19) , where ϕ p is the first eigenfunction of the p -Laplacian. In particular, we discover the formationof an S -shaped bifurcation diagram when p > q , see Figure 1.Prior to the rigorous description of main results, let us mention that various problemswith the ( p, q ) -Laplacian, whose motivation arises from the both mathematical and physicalpremises, are actively studied in the contemporary literature. Among physical origins of the ( p, q ) -Laplacian, one can think of it as a formal two-term Taylor approximation of more com-plex differential operators, see, e.g., [41] for the Zakharov equation describing in a simplifiedway long-wave oscillations of a plasma, [4] for a higher-dimensional generalization of the sine-Gordon equation which possesses soliton-type solutions, and [9] for an approximation of theelectrostatic Born-Infeld equation with a superposition of point charges. Let us also pointout a model in the theory of crystal growth containing the one-dimensional (1 , -Laplacianwhich was studied in [31]. Among mathematical origins, the ( p, q ) -Laplacian occurs, e.g., inthe procedure of elliptic regularization which consists in the inclusion of the regularizing term ε ∆ , ε ∈ R , in a nonlinear equation, with a view to obtain better properties of the augmentedequation, see, for instance, [1, 28]. Investigation of variational functionals with nonstandard ( p, q ) -growth conditions, mainly in connection with the Lavrentiev gap phenomenon, has beenperformed, e.g., in [16, 42]. Finally, we refer the interested reader to a nonexhaustive list ofworks [12, 13, 14, 27, 34, 35] for a development of the existence theory for various problemswith the ( p, q ) -Laplacian. Hereinafter, we denote the Sobolev space W ,r (Ω) shortly by W ,r , where r > . The standardnorm of the Lebesgue space L r (Ω) will be denoted by k · k r . A function u ∈ W ,p is called a(weak) solution of ( D α,β ) if the following equality is satisfied for any test function ϕ ∈ W ,p : Z Ω |∇ u | p − ∇ u ∇ ϕ dx + Z Ω |∇ u | q − ∇ u ∇ ϕ dx = α Z Ω | u | p − uϕ dx + β Z Ω | u | q − uϕ dx. (1.1)The energy functional E α,β : W ,p → R associated with ( D α,β ) is given by E α,β ( u ) = 1 p H α ( u ) + 1 q G β ( u ) , where H α ( u ) := k∇ u k pp − α k u k pp and G β ( u ) := k∇ u k qq − β k u k qq . Since p > q > , we have E α,β ∈ C ( W ,p , R ) , and hence weak solutions of ( D α,β ) are inone-to-one correspondence with critical points of E α,β .2 emark 1.1. Using the Moser iteration process (see, e.g., [29, Appendix A]), one can showthat any solution u of ( D α,β ) belongs to L ∞ (Ω) . Then, the regularity up to the boundarygiven by [24, Theorem 1] and [25, p. 320] ensures that u ∈ C ,γ (Ω) for some γ ∈ (0 , .Moreover, if u is a nonzero nonnegative solution, then the strong maximum principle and theboundary point lemma (see, e.g., [38, Theorems 5.4.1 and 5.5.1]) guarantee that u is positiveand belongs to int C (Ω) + := (cid:26) u ∈ C (Ω) : u ( x ) > for all x ∈ Ω , ∂u∂ν ( x ) < for all x ∈ ∂ Ω (cid:27) , the interior of the positive cone of C (Ω) . Here ν is the exterior unit normal vector to ∂ Ω .Finally, we denote by λ ( r ) the first eigenvalue of the r -Laplacian, i.e., λ ( r ) = inf (cid:26) k∇ u k rr k u k rr : u ∈ W ,r \ { } (cid:27) , and by ϕ r the corresponding first eigenfunction. Notice that ϕ r has a constant sign in Ω , andwe will assume, without loss of generality, that ϕ r > in Ω and k∇ ϕ r k r = 1 . Moreover, forsuch ϕ r we have ϕ r ∈ int C (Ω) + . Furthermore, since p > q , ϕ p cannot simultaneously be aneigenfunction of the q -Laplacian, see [7, Proposition 13]. Let us divide the ( α, β ) -plane into four open quadrants by the lines { λ ( p ) }× R and R ×{ λ ( q ) } (see Figures 1, 2). We recall several known facts about the existence, nonexistence, andmultiplicity of positive solutions of ( D α,β ) in these quadrants, as well as on their boundaries { λ ( p ) } × R and R × { λ ( q ) } . Proposition 1.2 ([5, Proposition 1] and [7, Proposition 13]) . Let α ≤ λ ( p ) and β ≤ λ ( q ) .Then ( D α,β ) has no nonzero solution. Proposition 1.3 ([5, Propositions 2 and 6] and [7, Remark 1]; see also [27, Lemma 2.2] for arelated result) . Let α < λ ( p ) and β > λ ( q ) . Then ( D α,β ) has at least one positive solution.Moreover, any nonzero solution u of ( D α,β ) satisfies E α,β ( u ) < . Furthermore, if α ≤ ,then the positive solution is unique. Proposition 1.4 ([5, Propositions 2 and 6]) . Let α > λ ( p ) and β < λ ( q ) . Then ( D α,β ) hasat least one positive solution. Moreover, any nonzero solution u of ( D α,β ) satisfies E α,β ( u ) > . In order to discuss the existence in the remaining quadrant ( λ ( p ) , + ∞ ) × ( λ ( q ) , + ∞ ) ,we introduce the threshold curve β ps ( α ) := sup { β ∈ R : ( D α,β ) has at least one positive solution } for α ≥ λ ( p ) . Define also the values α ∗ = k∇ ϕ q k pp k ϕ q k pp and β ∗ = k∇ ϕ p k qq k ϕ p k qq . It was proved in [5, Proposition 3] (see also [7, Remark 4]) that β ps ( α ) < + ∞ for any α >λ ( p ) , β ps ( λ ( p )) ≥ β ∗ , β ps ( · ) is continuous and nonincreasing on ( λ ( p ) , + ∞ ) , and β ps ( α ) = λ ( q ) for all α ≥ α ∗ . Moreover, α ∗ > λ ( p ) and β ∗ > λ ( q ) , see [7, Lemma 2.1].3 heorem 1.5 ([5, Theorem 2.2 and Proposition 4]) . Let α ∈ ( λ ( p ) , α ∗ ) and β ∈ ( −∞ , β ps ( α )] .Then ( D α,β ) has at least one positive solution. However, the properties of β ps ( α ) for α ∈ [ λ ( p ) , α ∗ ) are far from being completely un-derstood. In particular, the asymptotic behaviour of β ps ( α ) as α approaches λ ( p ) was sub-stantially unclear until the recent work [8], where the following results have been establishedby obtaining a nontrivial generalization of the classical Picone inequality [2] and using thegeneralized Picone inequalities [10, Proposition 2.9], [20, Lemma 1], and a radial symmetryresult of [11]. Theorem 1.6 ([8, Theorem 3.3]) . We have β ∗ ≤ β ps ( λ ( p )) < + ∞ . Moreover, ( D λ ( p ) ,β ) hasat least one positive solution if λ ( q ) < β < β ps ( λ ( p )) . Furthermore, if β ps ( λ ( p )) > β ∗ , then ( D λ ( p ) ,β ) has at least one positive solution if and only if λ ( q ) < β ≤ β ps ( λ ( p )) . Theorem 1.7 ([8, Theorem 3.2]) . Assume that one of the following assumptions is satisfied: (i) p ∈ I ( q ) , where I ( q ) := { p > q − s p + qs p − − ( p − q ) s + ( q − p + 1) ≥ for all s ≥ } ; (ii) p ≤ q + 1 and Ω is an N -ball.Then ( D λ ( p ) ,β ) has no positive solution for β > β ∗ , that is, β ps ( λ ( p )) = β ∗ . Moreover, if p < q + 1 and Ω is an N -ball, then ( D λ ( p ) ,β ) has no positive solution also for β = β ∗ . Remark 1.8.
We recall that q < p in Theorem 1.7 by default. The set I ( q ) is characterized in[8, Lemma 1.6]. In particular, it is known that for each q > there exists e p ∈ (max { , q } , q +1) such that [2 , e p ] ⊂ I ( q ) and ( e p, + ∞ ) ∩ I ( q ) = ∅ .Theorem 1.7 generates a natural question on whether β ps ( λ ( p )) > β ∗ if either the as-sumption (i) or (ii) of this theorem is violated. Nontriviality of this question is supported bythe fact that the behaviour of the energy functional E α,β at the point ( λ ( p ) , β ∗ ) cruciallydepends on the relation between p and q . Theorem 1.9 ([7, Theorem 2.6 (ii) and Remark 5]) . We have the following assertions: (i) If p < q , then inf W ,p E λ ( p ) ,β ∗ = −∞ . (ii) If p = 2 q , and ∂ Ω is connected when N ≥ , then inf W ,p E λ ( p ) ,β ∗ ∈ ( −∞ , . (iii) If p > q , and ∂ Ω is connected when N ≥ , then inf W ,p E λ ( p ) ,β ∗ ∈ ( −∞ , and theinfimum is attained by a positive solution of ( D λ ( p ) ,β ∗ ) . Let us remark that the connectedness of ∂ Ω is required in the proof of Theorem 1.9 (ii),(iii) due to the usage of the improved Poincaré inequality obtained in [17]. It is conjectured in[17, Section 3.1], however, that this assumption on ∂ Ω can be omitted for sufficiently regulardomains. To the best of our knowledge, this conjecture is still open.Theorem 1.9 suggests that not only the behaviour of E λ ( p ) ,β ∗ but also the structure ofthe solution set of ( D α,β ) in a neighbourhood of the point ( λ ( p ) , β ∗ ) is different in the cases p < q , p = 2 q , and p > q , which possibly affects the relation between β ps ( λ ( p )) and β ∗ .4ndeed, this turns out to be true, and the precise results will be formulated in Section 1.3below.A finer existence result in the quadrant ( λ ( p ) , + ∞ ) × ( λ ( q ) , + ∞ ) can be obtained if weintroduce the following family of critical values for α ≥ λ ( p ) : β ∗ ( α ) := inf (cid:26) k∇ u k qq k u k qq : u ∈ W ,p \ { } and H α ( u ) ≤ (cid:27) , (1.2)or, equivalently, β ∗ ( α ) = inf (cid:26) k∇ u k qq k u k qq : u ∈ W ,p \ { } and k∇ u k pp k u k pp ≤ α (cid:27) . It is known that β ∗ ( · ) is continuous and nonincreasing on [ λ ( p ) , + ∞ ) , β ∗ ( λ ( p )) = β ∗ >β ∗ ( α ) ≥ λ ( q ) for α > λ ( p ) , and β ∗ ( α ) > λ ( q ) if and only if α < α ∗ , see [7, Proposition 7]. Theorem 1.10 ([7, Theorem 2.7]) . Let λ ( p ) < α < α ∗ and λ ( q ) < β ≤ β ∗ ( α ) . Then ( D α,β ) has at least two positive solutions u and u such that E α,β ( u ) < , E α,β ( u ) > if β < β ∗ ( α ) , and E α,β ( u ) = 0 if β = β ∗ ( α ) . Moreover, u is the global least energy solution, andif β < β ∗ ( α ) , then u has the least energy among all solutions w of ( D α,β ) with E α,β ( w ) > . In particular, Theorem 1.10 yields β ∗ ( α ) ≤ β ps ( α ) for all α ≥ λ ( p ) . (1.3)Moreover, β ∗ ( α ) = β ps ( α ) = λ ( q ) for all α ≥ α ∗ . The most essential open question here waswhether the strict inequality in (1.3) holds. This issue is addressed in the present article, seethe following subsection.Finally, in accordance with the results described above, the only place on the ( α, β ) -plane where it remains to discuss the existence of positive solutions of ( D α,β ) is the interval [ α ∗ , + ∞ ) × { λ ( q ) } . It was proved in [5, Proposition 4 (ii)] that ( D α,λ ( q ) ) has no positivesolution whenever α > α ∗ . In fact, the proof of [5, Proposition 4 (ii)] can be slightly updatedin order to show that the nonexistence persists also in the case α = α ∗ . Indeed, it followsfrom the proof of [5, Proposition 4 (ii)] that if ( D α ∗ ,λ ( q ) ) possesses a positive solution u , then u = kϕ q for some k > . However, this is impossible in view of [7, Proposition 13]. Thus,thanks to Proposition 1.2 and Theorem 1.5, ( D α,λ ( q ) ) possesses a positive solution if and onlyif α ∈ ( λ ( p ) , α ∗ ) . For convenience, we introduce the following hypothesis:(H) p > q , and if N ≥ , then ∂ Ω is connected.Our first result is devoted to the relation between β ps ( α ) and β ∗ ( α ) . Theorem 1.11.
Let α ∈ [ λ ( p ) , α ∗ ) , and assume (H) if α = λ ( p ) . Then there exists ˜ β ( α ) >β ∗ ( α ) such that ( D α,β ) possesses a positive solution u for any β ∈ ( β ∗ ( α ) , ˜ β ( α )] . Moreover, u is a local minimum point of E α,β and E α,β ( u ) < . β α ∗ β ∗ β ps ( λ ( p )) λ ( q ) λ ( p ) β ps ( α ) β ∗ ( α ) α ∗ ( β ) Figure 1: The case p > q . The behaviour of β ∗ ( α ) , β ps ( α ) , α ∗ ( β ) , and alleged “minimal”bifurcation diagrams for the L ∞ -norms of positive solutions of ( D α,β ) with respect to β forseveral fixed α ’s. Light grey - two positive solutions, one of which is with positive energy andanother one is with negative energy; grey - two positive solutions with negative energy; darkgrey - three positive solutions with negative energy.The idea of the proof of Theorem 1.11 is based on the resent work [21], where the authorsobtained a local continuation of the branch of least energy solutions of an elliptic problem withindefinite nonlinearity using an original variational argument of a constrained minimizationtype.Theorem 1.11 implies, in particular, that β ∗ ( α ) < β ps ( α ) for all α ∈ ( λ ( p ) , α ∗ ) , and that β ∗ < β ps ( λ ( p )) provided the additional assumption (H) is satisfied. On the otherhand, we know from Theorem 1.7 that β ∗ = β ps ( λ ( p )) if either p ∈ I ( q ) or p ≤ q + 1 and Ω isan N -ball. Nevertheless, it remains unknown whether β ∗ = β ps ( λ ( p )) for all p ≤ q regardlessof assumptions on Ω . Moreover, we do not know whether ( D λ ( p ) ,β ps ( λ ( p )) ) possesses a positivesolution provided β ps ( λ ( p )) = β ∗ , except in the case discussed in Theorem 1.7, cf. Theorem1.5.Theorem 1.11 in combination with the behaviour of E α,β investigated in [7] allows to statethe following multiplicity results, among which Theorem 1.13 is perhaps the most surprisingsince it indicates the occurrence of an S -shaped bifurcation diagram in the case p > q , seeFigure 1. Theorem 1.12.
Let α ∈ [ λ ( p ) , α ∗ ) and β ∈ ( β ∗ ( α ) , β ps ( α )) , and assume (H) if α = λ ( p ) .Then ( D α,β ) has at least two positive solutions u and u satisfying E α,β ( u ) < and E α,β ( u ) < . β ∗ λ ( q ) β ps ( α ) β ∗ ( α ) λ ( p ) α ∗ α Figure 2: p, q as in Theorem 1.7. The behaviour of β ∗ ( α ) , β ps ( α ) , and alleged “minimal”bifurcation diagrams for the L ∞ -norms of positive solutions of ( D α,β ) with respect to β forseveral fixed α ’s. Light grey - two positive solutions, one of which is with positive energy andanother one is with negative energy; grey - two positive solutions with negative energy. Theorem 1.13.
Assume (H) . Then for every β ∈ ( β ∗ , β ps ( λ ( p ))] there exists α ∗ ( β ) ∈ (0 , λ ( p )) such that ( D α,β ) has at least three positive solutions for any α ∈ ( α ∗ ( β ) , λ ( p )) . Remark 1.14.
All three solutions obtained in Theorem 1.13 have negative energy, see Propo-sition 1.3.Let us recall that if α ≤ , then the positive solution of ( D α,β ) is unique (see Proposition1.3), and it was unclear (see [7, Remark 1]) whether a difficulty to extend the uniqueness to α ∈ (0 , λ ( p )) lies only in the limitation of the method of the original proof, or a multiplicity ofpositive solutions can actually occur. Theorem 1.13 answers this question in a nontrivial way.We emphasize that this multiplicity result does not depend on the domain, as it happens,e.g., in the case of superlinear problems of the type − ∆ q u = | u | p − u , cf. [32]. Moreover,there is no simple a priori intuition about such multiplicity based on the behaviour of fiberfunctions of E α,β , since these functions have at most one critical point which is the point ofglobal minimum, see Section 2. Finally, let us mention that the S -shaped bifurcation diagramindicated by Theorem 1.13 clarifies the shape of the bifurcation diagrams (A) or (B) in [22]obtained for the one-dimensional version of ( D λ,λ ). See Figure 3 for some numerical resultsin the one-dimensional case. Remark 1.15.
Properties of the family of critical points α ∗ ( β ) , such as the continuity, mono-tonicity, etc., are mostly unknown. We anticipate that the set of parameters α , β correspondingto the existence of three positive solutions of ( D α,β ) can be extended to a larger region asdepicted by the dashed line on Figure 1.The rest of the article is structured as follows. In Section 2, we introduce a few additionalnotations and provide an auxiliary lemma needed for the proof of Theorem 1.11 which we7 .2 0.4 0.6 0.8 1.00.20.40.60.81.01.2 Figure 3: Three positive solutions of the one-dimensional problem ( D α,β ) on the interval (0 , found by the shooting method with q = 2 , p = 6 , α = λ ( p ) − . and β = β ∗ + 0 . ,consult with [6, Appendix A] for explicit formulas for λ ( p ) and β ∗ .establish in Section 3. Section 4 provides auxiliary results needed to prove Theorems 1.12and 1.13. These theorems are established in Section 5. Appendix A contains a “ W ,p versus C local minimizers”-type result for general problems with the ( p, q ) -Laplacian, which we alsoapply in Section 5.
2. Auxiliary results I. The fibered functional J α,β Take any u ∈ W ,p satisfying H α ( u ) · G β ( u ) < and consider the fiber function t E α,β ( tu ) for t ≥ . It is not hard to observe that this function has a unique critical point t α,β ( u ) > given by t α,β ( u ) = (cid:18) − G β ( u ) H α ( u ) (cid:19) p − q = | G β ( u ) | p − q | H α ( u ) | p − q , (2.1)see [5, Proposition 6]. Moreover, J α,β ( u ) := E α,β ( t α,β ( u ) u ) = − sign ( H α ( u )) p − qpq | G β ( u ) | pp − q | H α ( u ) | qp − q . (2.2)In particular, if G β ( u ) < < H α ( u ) , then t α,β ( u ) is the point of global minimum of thefunction t E α,β ( tu ) , i.e., min t> E α,β ( tu ) = E α,β ( t α,β ( u ) u ) ≡ J α,β ( u ) = − p − qpq | G β ( u ) | pp − q H α ( u ) qp − q < . (2.3)The functional J α,β is called fibered functional [36], it is -homogeneous, and if u is a criticalpoint of J α,β satisfying H α ( u ) · G β ( u ) < , then t α,β ( u ) u is a critical point of E α,β .By N α,β we denote the Nehari manifold associated to E α,β , that is, N α,β = n v ∈ W ,p \ { } : h E ′ α,β ( v ) , v i = H α ( v ) + G β ( v ) = 0 o . Clearly, this set contains all nonzero critical points of E α,β . Notice that if we take any u ∈ W ,p satisfying H α ( u ) · G β ( u ) < , then t α,β ( u ) u ∈ N α,β , see [7, Proposition 10].8he following auxiliary result will be used in the proof of Theorem 1.11 in Section 3 below. Lemma 2.1.
Let α ∈ [ λ ( p ) , α ∗ ) , { β n } ⊂ [ β ∗ ( α ) , + ∞ ) be a sequence convergent to β ≥ β ∗ ( α ) ,and µ ∈ ( λ ( q ) , β ∗ ( α )) . Assume that a sequence { w n } ⊂ W ,p satisfies k∇ w n k p = 1 for all n ∈ N , and let w ∈ W ,p be such that { w n } converges weakly in W ,p and strongly in L p (Ω) to w as n → + ∞ . Assume, moreover, that G µ ( w n ) ≤ < H α ( w n ) for all n ∈ N , and − ∞ ≤ lim inf n → + ∞ J α,β n ( w n ) < . (2.4) Then w in Ω , and we have G β ( w ) < G µ ( w ) ≤ < H α ( w ) and − ∞ < J α,β ( w ) ≤ lim inf n → + ∞ J α,β n ( w n ) . Proof.
First we show that w in Ω . Suppose, by contradiction, that w ≡ in Ω . That is, k w n k p → and k w n k q → as n → + ∞ . In particular, we have H α ( w n ) = 1 − o (1) . Moreover,since G µ ( w n ) ≤ , we see that k∇ w n k q → , which yields G β n ( w n ) → , and, consequently, J α,β n ( w n ) → . However, this is a contradiction to (2.4), and hence w in Ω .By the weak lower semicontinuity, we readily get G µ ( w ) ≤ lim inf n → + ∞ G µ ( w n ) ≤ , which implies that k∇ w k qq k w k qq ≤ µ < β ∗ ( α ) . Due to the definition (1.2) of β ∗ ( α ) , we conclude that H α ( w ) > . On the other hand, byour assumptions, we have β > µ and β n > µ for all n , which implies that G β ( w ) < and G β n ( w n ) < for all n . Therefore, the weak lower semicontinuity of G β n and H α yields −∞ < J α,β ( w ) ≤ lim inf n → + ∞ J α,β n ( w n ) , which completes the proof.
3. Beyond β ∗ ( α ) . The proof of Theorem 1.11 In this section, we prove Theorem 1.11. Throughout the section, we assume α ∈ [ λ ( p ) , α ∗ ) to be fixed , and we require the hypothesis (H) if α = λ ( p ) .The proof of Theorem 1.11 will rely on the consideration of the following minimizationproblem: J ( β, µ ) := inf n J α,β ( u ) : u ∈ W ,p , G µ ( u ) < < H α ( u ) o , (3.1)where we assume β ≥ β ∗ ( α ) and µ ∈ ( λ ( q ) , β ∗ ( α )] , and J α,β is the fibered functional definedby (2.2). Notice that the index of G β presented in J α,β is, in general, different from the indexof G µ presented in the constraint. To the best of our knowledge, the idea of introduction ofsuch constraints was originated in the work [21].Let us discuss several general properties of (3.1). The admissible set for J ( β, µ ) isnonempty because µ > λ ( q ) and α < α ∗ yield G µ ( ϕ q ) < < H α ( ϕ q ) . Consequently, wealways have J ( β, µ ) ≤ J α,β ( ϕ q ) < , (3.2)9ince β ≥ β ∗ ( α ) > λ ( q ) . If we let β = µ , then J ( β, β ) translates to the usual minimiza-tion problem of finding the least energy solution to ( D α,β ) , see, e.g., [5, 7]. In particular, J ( β ∗ ( α ) , β ∗ ( α )) is attained, and if u ∗ is a corresponding minimizer, then t α,β ( u ∗ ) u ∗ is a solu-tion of ( D α,β ∗ ( α ) ), see Theorem 1.10 in the case λ ( p ) < α < α ∗ and Theorem 1.9 in the case α = λ ( p ) . Let us define µ = µ ( α ) := sup (cid:26) k∇ u ∗ k qq k u ∗ k qq : u ∗ is a minimizer of J ( β ∗ ( α ) , β ∗ ( α )) (cid:27) . (3.3) Proposition 3.1. µ < β ∗ ( α ) .Proof. It is clear that µ ≤ β ∗ ( α ) , since otherwise G β ∗ ( α ) ( u ∗ ) > for some minimizer u ∗ of J ( β ∗ ( α ) , β ∗ ( α )) , which is impossible, see (3.1) with β = µ = β ∗ ( α ) . Suppose, contrary to ourclaim, that µ = β ∗ ( α ) . That is, there exists a sequence of minimizers { u k } of J ( β ∗ ( α ) , β ∗ ( α )) such that k∇ u k k qq k u k k qq → β ∗ ( α ) . Since J α,β is 0-homogeneous, we may assume, without loss ofgenerality, that k∇ u k k p = 1 for each k . Thus, the latter convergence yields G β ∗ ( α ) ( u k ) → .On the other hand, since J α,β ∗ ( α ) ( u k ) = J ( β ∗ ( α ) , β ∗ ( α )) < by (3.2), we get from (2.3) that H α ( u k ) = (cid:18) p − qpq (cid:19) p − qq | G β ∗ ( α ) ( u k ) | pq ( −J ( β ∗ ( α ) , β ∗ ( α ))) p − qq for all k ∈ N . (3.4)Substituting (3.4) into (2.1), we deduce, in view of the default assumption p > q , that t α,β ∗ ( α ) ( u k ) → + ∞ . Moreover, by considering | u k | if necessary, we may assume that u k ≥ in Ω for all k . Recall now that t α,β ∗ ( α ) ( u k ) u k is a solution of ( D α,β ∗ ( α ) ), and hence (cid:10) H ′ α ( u k ) , ϕ (cid:11) + t α,β ∗ ( α ) ( u k ) q − p D G ′ β ∗ ( α ) ( u k ) , ϕ E = 0 for all ϕ ∈ W ,p , which implies that u k → ϕ p (strongly) in W ,p , up to a subsequence, and α = λ ( p ) , see[6, Lemma 3.3]. Therefore, if we fixed α > λ ( p ) , then we get a contradiction, and, con-sequently, the proposition follows. Assume that we fixed α = λ ( p ) . Notice that in thiscase we require (H). Considering the L -orthogonal decomposition u k = γ k ϕ p + v k , where γ k = k ϕ p k − R Ω u k ϕ p dx and R Ω v k ϕ p dx = 0 , we see that γ k → and k∇ v k k p → . Employingnow the improved Poincaré inequality from [17] along the same lines as in the proof of [7,Proposition 11] (see, more precisely, [7, pp. 1233-1234]), we deduce that J λ ( p ) ,β ∗ ( u k ) → ,which contradicts (3.2). Hence the proof is complete.In general, if J ( β, µ ) is attained, then the corresponding minimizer generates a solutionof ( D α,β ). We detail this fact as follows. Proposition 3.2.
Let β ≥ β ∗ ( α ) and assume that u ∈ W ,p is a minimizer of J ( β, µ ) forsome µ ∈ ( λ ( q ) , β ∗ ( α )] . Then t α,β ( u ) u is a local minimum point of E α,β and E α,β ( t α,β ( u ) u ) ≡ J α,β ( u ) = J ( β, µ ) < . Proof.
Suppose, by contradiction, that there exists a sequence { u n } convergent to ˜ u := t α,β ( u ) u in W ,p such that E α,β ( u n ) < E α,β (˜ u ) for all n ∈ N . u is a minimizer of J ( β, µ ) , we have G µ (˜ u ) < < H α (˜ u ) , and hence G β ( u n ) ≤ G µ ( u n ) < < H α ( u n ) for all sufficiently large n , which means that any such u n is an admissible function for J ( β, µ ) .But then, using (2.3), we get the following contradiction: E α,β (˜ u ) = J α,β ( u ) = J ( β, µ ) ≤ J α,β ( u n ) = E α,β ( t α,β ( u n ) u n ) ≤ E α,β ( u n ) < E α,β (˜ u ) for all sufficiently large n .Let us now discuss the existence of a minimizer of J ( β, µ ) required in Proposition 3.2. Lemma 3.3.
Let β ≥ β ∗ ( α ) and µ ∈ ( λ ( q ) , β ∗ ( α )) . Then there exists a nonnegative function u ∈ W ,p satisfying k∇ u k p = 1 such that G µ ( u ) ≤ < H α ( u ) and J α,β ( u ) ≤ J ( β, µ ) < . (3.5) Proof.
First, we recall that J ( β, µ ) < by (3.2). Let { u n } be a minimizing sequence for J ( β, µ ) . Since J α,β is -homogeneous and even, we can assume, without loss of generality,that k∇ u n k p = 1 and u n ≥ in Ω for all n ∈ N by considering | u n | if necessary. Therefore,there exists a nonnegative function u ∈ W ,p such that u n ⇀ u in W ,p and u n → u in L p (Ω) , up to an appropriate subsequence. Applying Lemma 2.1 (with β n = β ), we get (3.5).Using again the -homogeneity of J α,β , we can assume that k∇ u k p = 1 , which completes theproof.If the function u obtained in Lemma 3.3 satisfies G µ ( u ) < , then u is a minimizer of J ( β, µ ) . Consequently, Proposition 3.2 in combination with Remark 1.1 imply that t α,β ( u ) u is a positive solution of ( D α,β ). Thus, the proof of Theorem 1.11 reduces to the search of such β > β ∗ ( α ) and µ ∈ ( λ ( q ) , β ∗ ( α )) that G µ ( u ) < . The details are as follows. Proof of Theorem 1.11.
Let us fix any µ ∈ ( µ , β ∗ ( α )) , where µ is defined in (3.3) and µ <β ∗ ( α ) by Proposition 3.1. Denote by u = u ( β ) a normalized nonnegative function given byLemma 3.3. We are going to obtain the existence of ˜ β ( α ) > β ∗ ( α ) such that G µ ( u ( β )) < for any β ∈ ( β ∗ ( α ) , ˜ β ( α )) . Suppose, contrary to our claim, that there exists a sequence { β n } such that β n ց β ∗ ( α ) and G µ ( u ( β n )) = 0 for all n ∈ N . We will reach a contradiction byshowing that the corresponding sequence { u ( β n ) } converges in W ,p , up to a subsequence,to a minimizer of J ( β ∗ ( α ) , β ∗ ( α )) , which is impossible in view of the definition of µ .Since k∇ u ( β n ) k p = 1 for all n , there exists ¯ u ≥ such that u ( β n ) converges to ¯ u weaklyin W ,p and strongly in L p (Ω) and L q (Ω) , up to an appropriate subsequence. At the sametime, in view of (3.5), we have G µ ( u ( β n )) = 0 < H α ( u ( β n )) for all n ∈ N , and − ∞ ≤ lim inf n → + ∞ J α,β n ( u ( β n )) ≤ lim inf n → + ∞ J ( β n , µ ) < , (3.6)where the last inequality in (3.6) follows from the uniform bound (3.2). Consequently, applyingLemma 2.1 (with β = β ∗ ( α ) ) to the sequence { u ( β n ) } , we deduce that G β ∗ ( α ) (¯ u ) < G µ (¯ u ) ≤ < H α (¯ u ) (3.7)11nd J ( β ∗ ( α ) , β ∗ ( α )) ≤ J α,β ∗ ( α ) (¯ u ) ≤ lim inf n → + ∞ J α,β n ( u ( β n )) ≤ lim inf n → + ∞ J ( β n , µ ) , (3.8)where the first inequality in (3.8) follows from the fact that ¯ u is an admissible function for J ( β ∗ ( α ) , β ∗ ( α )) , see (3.7). On the other hand, noting that any minimizer u ∗ of J ( β ∗ ( α ) , β ∗ ( α )) satisfies G β n ( u ∗ ) < G µ ( u ∗ ) < < H α ( u ∗ ) by the definition of µ and the fact that β n > β ∗ ( α ) > µ > µ , we get J ( β n , µ ) ≤ J α,β n ( u ∗ ) = J α,β ∗ ( α ) ( u ∗ ) + o (1) = J ( β ∗ ( α ) , β ∗ ( α )) + o (1) . (3.9)Therefore, combining (3.8) with (3.9), we conclude that J ( β ∗ ( α ) , β ∗ ( α )) = J α,β ∗ ( α ) (¯ u ) , whichmeans that ¯ u is a minimizer of J ( β ∗ ( α ) , β ∗ ( α )) . Moreover, since (3.8) and (3.9) also imply J α,β ∗ ( α ) (¯ u ) = lim inf n → + ∞ J α,β n ( u ( β n )) , (3.10)we get u ( β n ) → ¯ u in W ,p , up to a subsequence. Indeed, if we suppose that there is no strongconvergence, then k∇ ¯ u k p < lim inf n → + ∞ k∇ u ( β n ) k p , and hence (0 < ) H α (¯ u ) < lim inf n → + ∞ H α ( u ( β n )) ,which implies a contradiction to the equality in (3.10). Finally, let us notice that the strongconvergence of { u ( β n ) } gives G µ (¯ u ) = 0 , see (3.7). However, this contradicts the definitionof µ and the fact that µ > µ .
4. Auxiliary results II. Mountain pass type arguments
In this section, we prepare several results related to the mountain pass theorem, which willbe used to prove Theorems 1.12 and 1.13 in Section 5 below. Since our aim is to find positive solutions of ( D α,β ), in the arguments of this section it will be convenient to consider the C -functional e E α,β ( u ) = 1 p e H α ( u ) + 1 q e G β ( u ) , u ∈ W ,p , where e H α ( u ) := k∇ u k pp − α k u + k pp and e G β ( u ) := k∇ u k q − β k u + k qq , and u + := max { u, } . The functional e E α,β differs from E α,β in that if u ∈ W ,p is an arbitrary critical point of e E α,β , then u is a nonnegative solution of ( D α,β ), which can be easily seen bytaking u − := max {− u, } as a test function. Moreover, u is a positive solution belonging to int C (Ω) + provided u , see Remark 1.1.Now we discuss the assumptions under which e E α,β satisfies the Palais–Smale condition. Lemma 4.1.
Let ( α, β ) = ( λ ( p ) , β ∗ ) . Then e E α,β satisfies the Palais–Smale condition.Proof. Let us take any Palais–Smale sequence { u n } for e E α,β . According to the ( S + ) -propertyof the operator − ∆ p − ∆ q (see, e.g., [6, Remark 3.5]), the desired Palais–Smale conditionfor e E α,β will follow if { u n } is bounded. Suppose, by contradiction, that k∇ u n k p → + ∞ as n → + ∞ , up to a subsequence. Considering the sequence of normalized functions v n := u n k∇ u n k p and arguing in much the same way as in [6, Lemma 3.3], we derive that { v n } converges in12 ,p , up to a subsequence, to some nonzero eigenfunction v of the p -Laplacian associated tothe eigenvalue α . Noting that o (1) k∇ ( u n ) − k p = D e E ′ α,β ( u n ) , − ( u n ) − E = k∇ ( u n ) − k pp + k∇ ( u n ) − k qq , we deduce that v ≥ in Ω . This yields α = λ ( p ) and v = ϕ p , since ϕ p is the only constant-sign eigenfunction of the p -Laplacian and we assumed that k∇ ϕ p k p = 1 . Thus, if α = λ ( p ) ,then we get a contradiction, and hence the Palais–Smale condition for e E α,β holds for any α = λ ( p ) and β ∈ R . On the other hand, in the case α = λ ( p ) and β = β ∗ , we get o (1) = 1 k∇ u n k qp (cid:16) p e E α,β ( u n ) − D e E ′ α,β ( u n ) , u n E(cid:17) = (cid:18) pq − (cid:19) e G β ( v n ) . This yields e G β ( ϕ p ) = G β ( ϕ p ) , which contradicts the assumption β = β ∗ .Before providing a mountain pass-type result, we give the following auxiliary lemma. Lemma 4.2.
Let u be a local minimum point of e E α,β such that inf u ∈N α,β E α,β ( u ) < e E α,β ( u ) < . (4.1) Then there exists a continuous path η ∈ C ([0 , , W ,p ) such that η (0) = u , e E α,β ( η (1)) < e E α,β ( u ) and max s ∈ [0 , e E α,β ( η ( s )) < . (4.2) Proof.
Noting that u ∈ int C (Ω) + (see Remark 1.1) and u ∈ N α,β , we get > e E α,β ( u ) = E α,β ( u ) = p − qpq G β ( u ) = − p − qpq H α ( u ) , and hence e G β ( u ) = G β ( u ) < < H α ( u ) = e H α ( u ) . According to (4.1), we can find v ∈ N α,β such that E α,β ( v ) < e E α,β ( u ) < . Moreover,since H α and G β are even, we may assume, by considering | v | if necessary, that v ≥ in Ω .Therefore, e G β ( v ) = G β ( v ) < < H α ( v ) = e H α ( v ) and e E α,β ( v ) < e E α,β ( u ) < . Let us consider the path ξ ( s ) = ((1 − s ) u q + sv q ) /q for s ∈ [0 , . The hidden convexity of ξ (see, e.g., [39, Lemma 2.4] or [10, Proposition 2.6]) implies that G β ( ξ ( s )) ≤ (1 − s ) G β ( u ) + sG β ( v ) ≤ max { G β ( u ) , G β ( v ) } < for all s ∈ [0 , . (4.3)Assume first that H α ( ξ ( s )) > for all s ∈ [0 , , and define the new path η ( s ) = t α,β ( ξ ( s )) ξ ( s ) , s ∈ [0 , , where t α,β ( ξ ( s )) is given by (2.1). Noting that t α,β ( ξ (0)) = t α,β ( ξ (1)) = 1 in viewof u , v ∈ N α,β , and that e E α,β ( η ( s )) = E α,β ( η ( s )) = J α,β ( ξ ( s )) = − p − qpq | G β ( ξ ( s )) | pp − q H α ( ξ ( s )) qp − q < for all s ∈ [0 ,
13y (2.3), we readily see that η satisfies (4.2).Recalling that H α ( ξ (0)) > and H α ( ξ (1)) > , assume now that there exists s ∈ (0 , such that H α ( ξ ( s )) = 0 . Without loss of generality, we may set s = inf { s ∈ (0 ,
1) : H α ( ξ ( s )) ≤ } , and so H α ( ξ ( s )) > for all s ∈ (0 , s ) . This implies that J α,β ( ξ ( s )) → −∞ as s ր s , thanksto (4.3). Thus, there exists some s ∈ (0 , s ) such that J α,β ( ξ ( s )) < e E α,β ( u ) . Considering the path η ( s ) = t α,β ( ξ ( s s )) ξ ( s s ) for s ∈ [0 , , we complete the proof. Theorem 4.3.
Let ( α, β ) = ( λ ( p ) , β ∗ ) . Assume that u is a local minimum point of e E α,β such that inf u ∈N α,β E α,β ( u ) < e E α,β ( u ) < . (4.4) Then there exists another critical point u of e E α,β satisfying e E α,β ( u ) ≤ e E α,β ( u ) < . (4.5) Proof.
Let η be a path given by Lemma 4.2. Since u is a local minimum point of e E α,β , thereexists r ∈ (0 , k∇ ( u − η (1)) k p ) such that e E α,β ( u ) ≤ e E α,β ( u ) < for every u ∈ B r ( u ) , where B r ( u ) = { u ∈ W ,p : k∇ ( u − u ) k p ≤ r } . Therefore, the generalized mountain passtheorem [37, Theorem 1] in combination with Lemma 4.1 implies that c := inf γ ∈ Γ max s ∈ [0 , e E α,β ( γ ( s )) ≥ e E α,β ( u ) is a critical level of e E α,β , and there exists a critical point u on the level c which is differentfrom u . Here Γ := n γ ∈ C ([0 , , W ,p ) : γ (0) = u , γ (1) = η (1) o . The properties (4.2) of the admissible path η yield c < , which gives (4.5).
5. Multiplicity. The proofs of Theorems 1.12 and 1.13
In this section, we prove Theorems 1.12 and 1.13 using the results of Section 4. A localminimum point of e E α,β will be obtained by the super- and subsolution method. Let usdenote, for brevity, f α,β ( u ) = α | u | p − u + β | u | q − u, and recall that a function u ∈ W ,p is called supersolution (resp. subsolution) of ( D α,β ) if u ≥ (resp. ≤ ) on ∂ Ω in the sense of traces and Z Ω |∇ u | p − ∇ u ∇ ϕ dx + Z Ω |∇ u | q − ∇ u ∇ ϕ dx ≥ Z Ω f α,β ( u ) ϕ dx ( resp. ≤ (5.1)14or any nonnegative ϕ ∈ W ,p . If, in addition, the strict inequality in (5.1) is satisfied for anynonnegative and nonzero ϕ , then u is called strict supersolution (resp. strict subsolution) of( D α,β ).Taking any v, w ∈ L ∞ (Ω) such that v ≤ w a.e. in Ω , we introduce the truncation f [ v,w ] α,β ( x, t ) = f α,β ( v ( x )) if t ≤ v ( x ) ,f α,β ( t ) if v ( x ) < t < w ( x ) ,f α,β ( w ( x )) if t ≥ w ( x ) , and define the corresponding C -functional E [ v,w ] α,β ( u ) = 1 p Z Ω |∇ u | p dx + 1 q Z Ω |∇ u | q dx − Z Ω Z u ( x )0 f [ v,w ] α,β ( x, t ) dt dx, u ∈ W ,p . If v and w are sub- and supersolutions of ( D α,β ), respectively, then critical points of E [ v,w ] α,β are solutions of ( D α,β ) and they belong to the ordered interval [ v, w ] , see, e.g., [5, Remark 2].We will make use of the following two lemmas. Lemma 5.1 ([5, Lemma 6 and Remark 2]) . Let α ∈ R and β > λ ( q ) , and let w ∈ int C (Ω) + be a positive supersolution of ( D α,β ) . Then inf W ,p E [0 ,w ] α,β < , the infimum is attained, andthe corresponding global minimum point u ∈ [0 , w ] satisfies ( D α,β ) and belongs to int C (Ω) + . Lemma 5.2.
Let α ≥ and β > λ ( q ) . Let w ∈ int C (Ω) + be a positive strict supersolutionof ( D α,β ) , that is, h E ′ α,β ( w ) , ϕ i > for any nonnegative and nonzero ϕ ∈ W ,p . (5.2) Let u ∈ int C (Ω) + be a global minimum point of E [0 ,w ] α,β given by Lemma 5.1. Then u ∈ (0 , w ) and u is a local minimum point of both E α,β and e E α,β in C (Ω) -topology.Proof. Noting that u ∈ (0 , w ] and ∂u∂ν , ∂w∂ν < on ∂ Ω , we will prove that u < w in Ω and ∂u∂ν > ∂w∂ν on ∂ Ω . (5.3)This fact directly implies the desired results. Indeed, if (5.3) holds, then w − u ∈ int C (Ω) + and hence, taking a sufficiently small κ > , we get u + v ∈ [0 , w ] for any v ∈ C (Ω) satisfying k v k C (Ω) < κ , whence E α,β ( u ) = e E α,β ( u ) = E [0 ,w ] α,β ( u ) = inf W ,p E [0 ,w ] α,β ≤ E [0 ,w ] α,β ( u + v ) = e E α,β ( u + v ) = E α,β ( u + v ) . To establish (5.3), we first show that u < w in a neighbourhood of ∂ Ω , and then we derivethat u < w in the remaining part of Ω . The details are as follows.For a sufficiently small δ > , we define Ω δ = { x ∈ Ω : dist ( x, ∂ Ω) < δ } . u, w ∈ int C (Ω) + , one can find ε, δ > such that |∇ ((1 − s ) u + sw ) | > ε in Ω δ for all s ∈ [0 , . Indeed, suppose, by contradiction, that for any n ∈ N there exist x n ∈ Ω /n and s n ∈ [0 , such that |∇ ((1 − s n ) u ( x n ) + s n w ( x n )) | ≤ n . Passing to appropriate subsequences,we get x n → x ∈ ∂ Ω , s n → s ∈ [0 , , and |∇ ((1 − s ) u ( x ) + s w ( x )) | = 0 . However, thiscontradicts the fact that ∂u∂ν ( x ) , ∂w∂ν ( x ) < .Let us denote, for short, A ( a ) := | a | p − a + | a | q − a for a ∈ R N , and define the linealization N × N -matrix A ( a ) as A ( a ) = | a | p − (cid:20) I + ( p − a ⊗ a | a | (cid:21) + | a | q − (cid:20) I + ( q − a ⊗ a | a | (cid:21) for a ∈ R N \ { } , where I is the identity matrix and ⊗ denotes the Kronecker product, see, e.g., [33, AppendixA.2]. Consider now v := w − u . Clearly, v ≥ in Ω . Recalling that w is a strict supersolutionof ( D α,β ), we subtract (1.1) from (5.2) and deduce, according to the mean value theorem, that v satisfies − div (cid:18)(cid:20)Z A ( ∇ ((1 − s ) u + sw )) ds (cid:21) ∇ v (cid:19) = − div ( A ( ∇ w ) − A ( ∇ u )) > f α,β ( w ) − f α,β ( u ) ≥ in Ω δ , (5.4)in the weak sense, where the last inequality follows from the fact that α and β are nonnegativeand w ≥ u > in Ω . Applying the estimates [33, (A.10)] to the matrix A , we get (cid:0) min { , p − }| a | p − + min { , q − }| a | q − (cid:1) | ξ | ≤ h A ( a ) ξ, ξ i R N ≤ (cid:0) max { , p − }| a | p − + max { , q − }| a | q − (cid:1) | ξ | (5.5)for any ξ ∈ R N and a ∈ R N \ { } . Here, for clarity, we denote by h· , ·i R N the usual scalarproduct in R N . Recalling that |∇ ((1 − s ) u + sw ) | > ε in Ω δ for all s ∈ [0 , , we employ theinequalities [33, (A.4) and (A.6)] to see that for any r > there exist C , C > such that C (cid:18) max s ∈ [0 , |∇ ((1 − s ) u + sw ) | (cid:19) r − ≤ Z |∇ ((1 − s ) u + sw ) | r − ds ≤ C (cid:18) max s ∈ [0 , |∇ ((1 − s ) u + sw ) | (cid:19) r − in Ω δ . (5.6)Thus, taking a = ∇ ((1 − s ) u + sw ) in (5.5) and using (5.6) with r = p and r = q , we concludethat there exist C , C > satisfying C | ξ | ≤ (cid:28)(cid:20)Z A ( ∇ ((1 − s ) u + sw )) ds (cid:21) ξ, ξ (cid:29) R N ≤ C | ξ | in Ω δ , for any ξ ∈ R N . That is, the differential operator in (5.4) is uniformly elliptic in Ω δ . Therefore, in view of thestrict inequality in (5.4), the strong maximum principle yields v > in Ω δ and ∂v∂ν < on ∂ Ω .Consequently, u < w in Ω δ and ∂u∂ν > ∂w∂ν on ∂ Ω .16et us now fix some δ ′ ∈ (0 , δ ) and a sufficiently small C > such that u + C ≤ w on ∂ Ω δ ′ ∩ Ω . Denoting z = u + C , we see that Z Ω |∇ z | p − ∇ z ∇ ϕ dx + Z Ω |∇ z | q − ∇ z ∇ ϕ dx = Z Ω f α,β ( u ) ϕ dx for any ϕ ∈ W ,p . (5.7)Therefore, subtracting (5.2) from (5.7) and taking ϕ = max { z − w, } in Ω \ Ω δ ′ and ϕ = 0 in Ω δ ′ , we derive that ≤ Z { z>w }∩ (Ω \ Ω δ ′ ) (cid:0) |∇ z | p − ∇ z − |∇ w | p − ∇ w (cid:1) ( ∇ z − ∇ w ) dx + Z { z>w }∩ (Ω \ Ω δ ′ ) (cid:0) |∇ z | q − ∇ z − |∇ w | q − ∇ w (cid:1) ( ∇ z − ∇ w ) dx ≤ Z { z>w }∩ (Ω \ Ω δ ′ ) ( f α,β ( u ) − f α,β ( w )) ( z − w ) dx ≤ , which implies that { z > w } = ∅ in Ω \ Ω δ ′ . Thus, u + C ≤ w and, consequently, u < w in Ω \ Ω δ ′ . Recalling that u < w in Ω δ , we conclude that u ∈ (0 , w ) in Ω . Thus, (5.3) is satisfied,which completes the proof. Fix any α ∈ [ λ ( p ) , α ∗ ) and β ∈ ( β ∗ ( α ) , β ps ( α )) . Choosing an arbitrary β ′ ∈ ( β, β ps ( α )] ,we denote by w ∈ int C (Ω) + a positive solution of ( D α,β ′ ), see Theorem 1.5 in the case α > λ ( p ) and Theorem 1.6 in the case α = λ ( p ) for the existence result. Clearly, w is astrict supersolution of ( D α,β ). Hence, thanks to Lemma 5.1, we can find a global minimumpoint u ∈ int C (Ω) + of E [0 ,w ] α,β such that E α,β ( u ) = E [0 ,w ] α,β ( u ) < , and u is a positivesolution of ( D α,β ). Moreover, according to Lemma 5.2, u is a local minimum point of e E α,β in C (Ω) -topology. Therefore, applying Theorem A.1 with f ( x, t ) = αt p − + βt q − , we see that u is a local minimum point of e E α,β in W ,p .On the other hand, it was shown in [7, Theorem 2.5] that inf u ∈N α,β E α,β ( u ) = −∞ provided β > β ∗ ( α ) . Consequently, (4.4) holds, whence Theorem 4.3 yields the existence of the secondpositive solution u of ( D α,β ) which satisfies (4.5). Fix any β ∈ ( β ∗ , β ps ( λ ( p ))] and denote by w ∈ int C (Ω) + a positive solution of ( D λ ( p ) ,β ),see Theorem 1.6 for the existence of w . Evidently, w is a strict supersolution of ( D α,β ) forany α ∈ [0 , λ ( p )) . Therefore, arguing as in the proof of Theorem 1.12 above, we can find alocal minimum point u = u ( α ) ∈ int C (Ω) + of e E α,β in W ,p such that u ( α ) ∈ (0 , w ) and e E α,β ( u ( α )) = E α,β ( u ( α )) < for any α ∈ [0 , λ ( p )) . (5.8)Thus, u ( α ) is the first positive solution of ( D α,β ). Moreover, in view of the uniform L ∞ -boundof u ( α ) in (5.8), we get inf { E α,β ( u ( α )) : α ∈ [0 , λ ( p )) } > −∞ . (5.9)17et u = u ( α ) ∈ int C (Ω) + be a global minimum point of E α,β for α < λ ( p ) obtainedin [7, Proposition 1]. It is proved in [7, Proposition 2 (i)] that E α,β ( u ( α )) → −∞ , k u ( α ) k p → + ∞ , and u ( α ) k u ( α ) k p → ϕ p k ϕ p k p in W ,p (5.10)as α ր λ ( p ) . Comparing (5.9) and (5.10), we derive the existence of α ∗ ( β ) ∈ [0 , λ ( p )) suchthat E α,β ( u ( α )) < E α,β ( u ( α )) for any α ∈ ( α ∗ ( β ) , λ ( p )) . (5.11)Hence, u ( α ) = u ( α ) whenever α ∈ ( α ∗ ( β ) , λ ( p )) . Moreover, we note that, in fact, α ∗ ( β ) ∈ (0 , λ ( p )) due to the uniqueness result in Proposition 1.3. On the other hand, in view of (5.11),Theorem 4.3 provides us with the existence of the third positive solution u ( α ) of ( D α,β ) forany α ∈ ( α ∗ ( β ) , λ ( p )) , and u ( α ) is different from u ( α ) and u ( α ) . A. W ,p versus C local minimizers Let f : Ω × R → R be any Carathéodory function and let F ( x, u ) = R u f ( x, v ) dv be theprimitive of f . Along this section, we assume that f satisfies the following subcritical growthcondition:(G) There exist C > and r ∈ [1 , p ∗ ) such that | f ( x, t ) | ≤ C (1 + | t | r − ) for every t ∈ R and a.e. x ∈ Ω , where p ∗ = pNN − p if N > p , and p ∗ = + ∞ if N ≤ p .It is well known that the functional I ( u ) = 1 p Z Ω |∇ u | p dx + 1 q Z Ω |∇ u | q dx − Z Ω F ( x, u ) dx, u ∈ W ,p , is weakly lower semicontinuous and of class C under the assumption (G).The following result can be obtained in much the same way as [19, Theorem 1.2] or [29,Theorem 23], see also [23] for a generalization. For the convenience of the reader we sketchits proof based on [29, Theorem 23]. Theorem A.1.
Let u ∈ W ,p be a local minimum point of I in C (Ω) -topology, namely,there exists ε > such that I ( u ) ≤ I ( u + h ) for any h ∈ C (Ω) satisfying k h k C (Ω) < ε. (A.1) Then u is also a local minimum point of I in W ,p -topology.Proof. Since h I ′ ( u ) , h i = 0 for every h ∈ C (Ω) and since C (Ω) is dense in W ,p , we deducethat u is a critical point of I . That is, − ∆ p u − ∆ q u = f ( x, u ) in Ω , (A.2)18n the weak sense. Moreover, one can show that u ∈ C ,ν (Ω) for some ν ∈ (0 , , cf. Remark1.1 or [26, Section 2.4].Suppose, by contradiction, that u is not a local minimum point of I in W ,p -topology.Then for any sufficiently small ε > we have m ε := inf n I ( u + h ) : h ∈ e B ε (0) o < I ( u ) , where e B ε (0) = { v ∈ W ,p : k v k r ≤ ε } and r ∈ [1 , p ∗ ) is given in the assumption (G). Let { h n } be a minimizing sequence for m ε with a fixed ε > . Thanks to (G), { h n } is bounded in W ,p .Therefore, m ε is attained by some h ε ∈ e B ε (0) , since I is weakly lower semicontinuous on W ,p and e B ε (0) is weakly closed in W ,p . Then, due to the Lagrange multipliers rule, there exists λ ε ≤ such that − ∆ p ( u + h ε ) − ∆ q ( u + h ε ) = f ( x, u + h ε ) + λ ε | h ε | r − h ε in Ω . (A.3)Denoting now e A ( x, y ) = |∇ u ( x ) + y | p − ( ∇ u ( x ) + y ) + |∇ u ( x ) + y | q − ( ∇ u ( x ) + y ) − |∇ u ( x ) | p − ∇ u ( x ) − |∇ u ( x ) | q − ∇ u ( x ) , we subtract (A.2) from (A.3) and get − div e A ( x, ∇ h ε ) = f ( x, u + h ε ) − f ( x, u ) + λ ε | h ε | r − h ε in Ω . Recalling that λ ε ≤ and using the Moser iteration method (see, e.g., [29, Theorem C]), wecan find M > independent of ε such that k h ε k ∞ ≤ M for every ε > . Then, applying theregularity result of [25] to the solution u + h ε of (A.3), we obtain that u + h ε ∈ C (Ω) , andso h ε ∈ C (Ω) for every ε > .Finally, it can be shown as in [29, Theorem 23] that there exists d > such that | λ ε | h ε ( x ) | r − h ε ( x ) | ≤ d for every x ∈ Ω and ε > . This implies that f ( x, u ) + λ ε | h ε ( x ) | r − h ε ( x ) is bounded on Ω × [ − M − k u k ∞ , M + k u k ∞ ] uniformly in ε > . Thus, applying again the regularity result of [25] to the solution u + h ε of (A.3), we deduce the existence of θ ∈ (0 , and M > , both independent of ε , such that u + h ε ∈ C ,θ (Ω) and k u + h ε k C ,θ (Ω) ≤ M for every ε > . Since C ,θ (Ω) is embeddedcompactly into C (Ω) , we infer that u + h ε → u as ε ց in C (Ω) by noting that h ε → in L r (Ω) as ε ց . Consequently, we get the following contradiction between (A.1) and (A.2): I ( u + h ε ) = m ε < I ( u ) ≤ I ( u + h ε ) for all sufficiently small ε > . Acknowledgements
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