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Anisotropic double-phase problems with indefinite potential: multiplicity of solutions

We consider an anisotropic double-phase problem plus an indefinite potential. The reaction is superlinear. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we prove a multiplicity theorem producing five nontrivial smooth solutions, all with sign information and ordered. In this process we also prove two results of independent interest, namely a maximum principle for anisotropic double-phase problems and a strong comparison principle for such solutions.


Introduction and origin of double-phase problems
⊆ R N be a bounded domain with a C 2 -boundary ∂ . In this paper we deal with the following anisotropic double phase Dirichlet problem In this problem, we assume that p, q ∈ C 1 ( ) and 1 < q − ≤ q(z) ≤ q + < p − ≤ p(z) ≤ p + < p * (z), where p * (z) = N p(z) N − p(z) if p + < N and +∞ otherwise. The potential function ξ ∈ L ∞ ( ) is sign-changing and so the differential operator (left-hand side) of problem (1) is not coercive. The reaction f (z, x) is a Carathéodory function (that is, for all x ∈ R the mapping z → f (z, x) is measurable and for a.a. z ∈ the function x → f (z, x) is continuous) which exhibits ( p + − 1)-superlinear growth near ±∞, but without satisfying the Ambrosetti-Rabinowitz condition (the AR-condition). Using variational tools from the critical point theory, together with truncation, perturbation and comparison techniques and critical groups, we show that the problem has at least five nontrivial smooth solutions, all with sign information and ordered.
The energy functional associated to problem (1) is a double-phase variational integral, according to the terminology of Marcellini and Mingione. Problems with unbalanced growth have been studied for the first time by Ball [4,5] in relationship with patterns arising in nonlinear elasticity. More precisely, if is a bounded domain in R N , u : → R N is the displacement and if Du is the N × N matrix of the deformation gradient, then Ball studied the total energy, which can be represented by an integral of the type where the energy function f = f (x, ξ) : × R N ×N → R is quasiconvex with respect to ξ . One of the simplest examples considered by Ball is given by functions f of the type where det ξ is the determinant of the N × N matrix ξ , and g, h are nonnegative convex functions, which satisfy the growth conditions where c 1 is a positive constant and 1 < p < N . The condition p ≤ N is necessary to study the existence of equilibrium solutions with cavities, that is, minima of the integral (2) that are discontinuous at one point where a cavity forms; in fact, every u with finite energy belongs to the Sobolev space W 1, p ( , R N ), and thus it is a continuous function if p > N .
The mathematical analysis of double-phase integral functionals has been initiated by Marcellini [20,21]. Marcellini considered continuous functions f = f (x, u) with unbalanced growth that satisfy where c 1 , c 2 are positive constants and 1 ≤ q ≤ p. These contributions are in relationship with the works of Zhikov [37,38], in order to 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. These functionals revealed to be important also in the study of duality theory and in the context of the Lavrentiev phenomenon. In particular, Zhikov considered the following three model functional in relation to the Lavrentiev phenomenon: The functional M is well-known and there is a loss of ellipticity on the set {x ∈ ; c(x) = 0}. This functional has been studied at length in the context of equations involving Muckenhoupt weights. The functional V has also been the object of intensive interest nowadays and a huge literature was developed on it. The energy functional defined by V was used to build models for strongly anisotropic materials: in a material made of different components, the exponent p(x) dictates the geometry of a composite that changes its hardening exponent according to the point. The functional P p,q defined in (3) appears as an upgraded version of V. Again, in this case, the modulating coefficient a(x) dictates the geometry of the composite made by two differential materials, with hardening exponents p and q, respectively. 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 continued in a series of remarkable papers by Mingione et al. [6,7,9].
This work continues the recent paper by Papageorgiou, Rȃdulescu & Repovš [26], where the authors consider parametric equations driven by the p(z)-Laplacian plus an indefinite potential term. In the reaction there are the competing effects of a parametric concave term and of a superlinear (convex) perturbation ("concave-convex" problem). The authors focus on positive solutions and they prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter λ > 0 varies. We also mention the work of Papageorgiou & Vetro [28], who also deal with anisotropic double phase problems with no potential term (that is, ξ ≡ 0) and with a superlinear reaction that has a different geometry near zero. They prove a multiplicity theorem producing three nontrivial solutions. However, they do not prove the existence of nodal solutions. Finally, we mention the work of Gasiński & Papageorgiou [14] on superlinear Neumann problems driven by the p(z)-Lapacian. Other anisotropic boundary value problems (including double phase problems) can be found in the book of Rȃdulescu & Repovš [31] and in the papers of Bahrouni, Rȃdulescu & Repovš [2,3], Cencelj, Rȃdulescu & Repovš [8], Papageorgiou, Vetro and Vetro [29], Ragusa and Tachikawa [30], Vetro and Vetro [34], and Zhang & Rȃdulescu [36].
The features of this paper are the following: (i) we are concerned with an anisotropic model with double-phase, namely the problem is driven by two differential operators with variable growth; (ii) we develop a refined mathematical analysis (that combines variational and topological methods) in order to study multiplicity properties of solutions; (iii) we establish both a maximum principle for anisotropic double-phase problems and a strong comparison principle for solutions of anisotropic PDEs with unbalanced growth.

Auxiliary results and hypotheses
The study of anisotropic boundary value problems uses variable exponent Lebesgue and Sobolev spaces. A comprehensive presentation of the theory of such spaces can be found in the book of Diening, Harjulehto, Hästo & Ruzička [10]. Let Given p ∈ L ∞ 1 ( ), we define p − = essinf p and p + = esssup p.
We also let M( ) = {u : → R measurable}. We identify two such functions which differ on a Lebesgue null set.
Given p ∈ L ∞ 1 ( ), we define the variable exponent Lebesgue space L p(z) ( ) by This space is furnished with the so-called Luxemburg norm defined by Using these variable exponent Lebesgue spaces, we can define the corresponding variable exponent Sobolev spaces by The norm of this space is given by When p − > 1, then the spaces L p(z) ( ), ( ) are separable and uniformly convex (thus, reflexive too).
The critical Sobolev exponent is defined by Let p, p ∈ L ∞ 1 ( ) and assume that 1 p(z) + 1 p (z) = 1 for a.a. z ∈ . We have L p(z) ( ) * = L p (z) ( ) and the following Hölder-type inequality holds When p ∈ C 1 ( ), the Poincaré inequality holds for the space W 1, p(z) 0 ( ), namely there exists C * > 0 such that The following modular functions are important in the study of these anisotropic spaces: The next propositions reveal the close relation between these modular functions and the norms of the spaces.

Proposition 2
If p ∈ C 1 ( ), then the following properties hold: For p ∈ C 1 ( ), we have Then we consider the operator A p : W Our hypotheses on the exponents p, q and the potential function ξ are: For every x ∈ R, we set x ± = max{±x, 0} and then given u ∈ W 1, p(z) 0 ( ) we define u ± (z) = u(z) ± for all z ∈ . We know that ( ) is said to be "downward directed" (resp., "upward directed"), if for all u 1 , u 2 ∈ S, we can find u ∈ S such that u ≤ u 1 , u ≤ u 2 (resp., for all Let X be a Banach space and ϕ ∈ C 1 (X , R). We say that ϕ(·) satisfies the "Ccondition", if the following property holds: admits a strongly convergent subsequence" .
For ϕ(·) we define K ϕ = {u ∈ X : ϕ (u) = 0} (critical set of ϕ), the kth relative singular homology group with integer coefficients. Then for u ∈ K ϕ isolated and c = ϕ(u), we define the "kth critical group" of ϕ(·) at u, by with U a neighborhood of u, such that K ϕ ∩ ϕ c ∩ U = {u}. The excision property of singular homology implies that this definition is independent of the isolating neighborhood U .
The regularity theory for anisotropic problems will lead us to the Banach space This is an ordered Banach space with positive (order) cone This cone has a nonempty interior given by with n(·) being the outward unit normal on ∂ .
In what follows, we denote by · the norm of the Sobolev space W 1, p(z) 0 ( ). On account of the Poincaré inequality, we have Next, we will prove two auxiliary results which are actually of independent interest. The first is a strong maximum principle for anisotropic double phase problems. Our result complements the analogous result by Zhang [35]. His conditions on the differential operator do not cover double phase problems (see conditions (3)-(7) in [35]).

Proposition 4 If hypotheses H
Arguing by contradiction, suppose we can find z 1 , z 2 ∈ and an open ball We introduce the following items We can easily check that Choose ρ > 0 small so that m ρ < 1 (see (5)) and To simplify things, we may take without any loss of generality z 2 = 0. We set . (6) and recall that q(·) < p(·)) is a lower solution of (4) on 1 .
Note that y ≤ u on ∂ 1 . So, by Lemma 2.3 of Zhang [35] we have that Hence we have which contradicts (5). Therefore we infer that Next, let z 1 ∈ ∂ and let ρ > 0 be small. We set z 2 = z 1 − 2ρn(z 1 ) and have .
In what follows, we denote by D + the following open cone in C 1 ( ): Then a i j ∈ W 1,∞ ( ) and by the mean value theorem we have (see also Guedda & Véron [16]). Suppose that there exists z 0 ∈ such that u(z 0 ) = v(z 0 ). Hence y(z 0 ) = 0. From our hypotheses and since the function (z, x) → |x| p(z)−2 x is uniformly continuous on × R, we see that we can find δ > 0 small such that (see Theorem 4 of Vazquez [33]), a contradiction since y(z 0 ) = 0. So, we have The proof is now complete.
Now we introduce the hypotheses on the reaction f (z, x) .
(v) we can find C 0 ,Ĉ > 0 such that We point out that we do not use the AR-condition, which is common in the literature when dealing with superlinear equations. Instead we use the quasimonotonicity hypothesis H (iii) on β(z, ·). This assumption is a slight generalization of the condition used by Li & Yang [19]. Similar conditions were used by Mugnai & Papageorgiou [22] (isotropic problems) and by Papageorgiou, Rȃdulescu & Repovš [26,27], Papageorgiou & Vetro [28] (anisotropic problems). With this condition we incorporate in our framework also superlinear functions with "slower" growth near ±∞ which fail to satisfy the AR-condition. For example, consider the function with the exponents p, q ∈ C 1 ( )

. This function satisfies hypotheses
H but fails to satisfy the AR-condition (see [1]). Hypotheses H (iv), (v) imply that f (z, ·) near zero has a kind of oscillatory behavior. Finally, we mention that another superlinearity condition for anisotropic equations was used by Gasiński & Papageorgiou [14].

Constant sign solutions
In this section we produce constant sign solutions.
We first produce two constant solutions. One solution is positive in the order interval [0, C 0 ] and the other solution is negative in the order interval [−Ĉ, 0]. To produce these two solutions, we do not need the hypotheses concerning the asymptotic behavior of f (z, ·) (hypotheses H (ii), (iii)).

Proposition 6 If hypotheses H 0 and H (i), (iv), (v) hold, then problem (1) admits two constant sign solutions
Proof First we produce the positive solution.
For the negative solution, we consider the following truncation perturbation of This is a Carathéodory function. We setF − (z, x) = x 0f − (z, s)ds and consider ( ). Working as above, using this timeφ − (·), we produce a negative solution The proof is now complete.
By introducing an extra mild condition on f (z, ·), we can improve the conclusion of the previous proposition. With this stronger conclusion, we will be able to produce in the sequel additional constant sign solutions.
The new conditions on the reaction f (z, x) are the following: H : f : ×R → R is a Carathéodory function such that f (z, 0) = 0 for a.a. z ∈ , hypotheses H (i) → (v) are the same as the corresponding hypotheses H (i) → (v) and (vi) for every ρ > 0, there existsξ ρ > 0 such that for a.a. z ∈ , the function Using this perturbed monotonicity condition on f (z, ·), we obtain the following improved version of Proposition 6.

Proposition 7
If hypotheses H 0 , H hold, then problem (1) admits two constant sign solutions Proof From Proposition 6, we already have two solutions Let ρ = C 0 and letξ ρ > 0 be as postulated by hypothesis H (vi). Clearly we can always haveξ ρ > ξ ∞ . Then (15) and hypothesis H (vi)) We will use these two solutions u 0 ∈ int C + and v 0 ∈ −int C + , in order to produce two more constant sign smooth solutions localized with respect to u 0 and v 0 respectively.

Proposition 8 If hypotheses H 0 , H hold, then problem (1) admits two more constant sign solutionsû
Proof Let u 0 ∈ int C + and v 0 ∈ −int C + be the two constant sign solutions from Proposition 6. From Proposition 7 we have First we will produce the second positive solution. To this end, we introduce the following truncation perturbation of f (z, ·): This is a Carathéodory function. We set G + (z, x) = From (19) we have In (20) for some C 3 > 0, all n ∈ N (see (15)), In (20) we choose h = u + n ∈ W 1, p(z) 0 ( ) and using (17), we have From (18) and (21) we have for some C 5 > 0, all n ∈ N (see (17)), We add (22) and (23) and obtain for some C 6 < 0, all n ∈ N. Using (24) we will show that {u + n } n≥1 ⊆ W 1, p(z) 0 ( ) is bounded. Arguing by contradiction, we assume that at least for a subsequence we have Let y n = u + n u + n for n ∈ N. Then y n = 1, y n ≥ 0 for all N and so we may assume that y n w → y in W 1, p(z) 0 ( ) and y n → y in L r ( ) as n → ∞, y ≥ 0.
From (26) it follows that On account of (25), we see that we can find n 0 ∈ N such that From (31) and (34), we have (since k > 1, y n = 1 and by Proposition 2) ≥ 1 2 p + k for all n ≥ n 1 ≥ n 0 (see (33)).
We may assume that In (20) we choose h = u n − u ∈ W 1, p(z) 0 ( ), pass to the limit as n → ∞ and use (39). Then This proves Claim 1. On account of hypothesis H (ii), for every u ∈ int C + , we have that (40) Using the anisotropic regularity theory (see Fan [11]) we deduce that u ∈ int C + . This proves Claim 2.
Recall that u 0 (z) < C 0 for all z ∈ . On account of Claim 2, we may assume that or otherwise we already have a second positive solution bigger than u 0 (see (17)) and so we are done. We consider the following truncation of g + (z, ·) This is a Carathéodory function. We setĜ + (z, x) = x 0ĝ + (z, s)ds and consider ( ). From (43) and since ϑ > ξ ∞ , we see thatˆ + (·) is coercive. Also it is sequentially weakly lower semicontinuous. So, we can findũ 0 ∈ W 1, p(z) 0 ( ) such that As in the proof of Claim 2, we show that So, we have proved that u ∈ [u 0 , C 0 ]. From this and the anisotropic regularity theory (see Fan [11]), we conclude that Kˆ + ⊆ [u 0 , C 0 ] ∩ int C + . This proves Claim 3.
Then from (42) and (45) it follows that Kˆ + = {u 0 }. Hence from (44) we have that u 0 = u 0 and since u 0 ∈ int C 1 0 ( ) [0, C 0 ] (see Proposition 7), from (45) we infer that On account of Claim 2, we may assume that Otherwise we already have an infinity of positive smooth solutions bigger than u 0 and so we are done.
To produce the second negative solution, we argue similarly starting from the Carathéodory function The proof is now complete.
We introduce the following sets S + = set of positive solutions of problem (1), S − = set of negative solutions of problem (1).
We already know that Moreover, S + is downward directed and S − is upward directed (see Papageorgiou, Rȃdulescu & Repovš [23]). We will show that there exist extremal constant sign solutions, that is, a smallest positive solution and a biggest negative solution. In the next section, we will use these extremal constant sign solutions in order to produce a nodal (sign-changing) solution.

Proposition 9
If hypotheses H 0 , H hold, then there exist u * ∈ S + and v * ∈ S − such that Proof Invoking Lemma 3.10 of Hu & Papageorgiou [17, p. 178], we can find a decreasing sequence {u n } n≥1 ⊆ S + such that inf n≥1 u n = inf S + .

We have
If in (50) we choose h = u n ∈ W 1, p(z) 0 ( ) and use (51) and hypothesis H (i), we see that So, we may assume that In (50) we choose h = u n − u * ∈ W 1, p(z) 0 ( ), pass to the limit as n → ∞ and use (53). Then, as before (see the proof of Proposition 8, Claim 1), we obtain lim sup We need to show that u * = 0. On account of hypotheses H (i), (iv), given any η > 0, we can find C 14 = C 14 (η) > 0 such that We consider the following auxiliary anisotropic Dirichlet problem: Claim 1: Problem (56) admits a unique positive solution u ∈ int C + and since the problem is odd, then v = −u ∈ −int C + is the unique negative solution of (56).
First we show the existence of a positive solution. So, we consider the C 1 -functional Since q − ≤ q(z) < p(z) < r for all z ∈ , we see that τ + (·) is coercive. Also it is sequentially weakly lower semicontinuous. So, we can find u ∈ W 1, p(z) 0 ( ) such that Fix u ∈ int C + . For t ∈ (0, 1), we have Recall that η > 0 is arbitrary. So, choosing η > ρ q (Du) ρ q − (u) and t ∈ (0, 1) even smaller if necessary, we have that So, u is a positive solution of (56) and from the anisotropic regularity theory and Proposition 4, we have u ∈ int C + .
From (58) Let h = u q − −ũ q − . Then for |t| ≤ 1 small we have Hence the functional j(·) is Gâteaux differentiable at u q and atũ q in the direction h. Moreover, on account of the convexity of j(·) we obtain that j (·) is monotone.
For the biggest negative solution the reasoning is similar. In this case, since S − is upward directed, we can find {v n } n≥1 ⊆ S − increasing such that Then working as above, we obtain v * ∈ W The proof is now complete.

Nodal solutions
In this section, using the extremal constant sign solutions from Proposition 9, we will obtain a nodal (sign changing) solution.

Proposition 10
If hypotheses H 0 , H hold, then problem (1) admits a nodal solution with u * and v * being the two extremal constant sign solutions from Proposition 9.
From (74) we have Comparing (75) and (76), we obtain a contradiction. Therefore relation (73) is true. From (65) we see that we may assume that Kφ is finite. Otherwise we already have an infinity of nodal solutions (due to the extremality of u * and v * ). So, 0 ∈ K ϕ is isolated (recall that K ϕ | [v * ,u * ] = Kφ| [v * ,u * ] ) and so we can have ρ ∈ (0, 1) small On account of (73), this deformation is well defined and shows that ϕ 0 ∩ B ρ is contractible.
From the equation in (83) and Theorem 4.1 of Fan & Zhao [12], we know that u n ∈ L ∞ ( ) and u n ∞ ≤ C 21 for some C 21 > 0, all n ∈ N.
The proof is now complete.
So, summarizing we can state the following multiplicity theorem for problem (1).

Remark 2
We emphasize that in the above multiplicity theorem we provide sign information for the solutions and moreover, the solutions are linearly ordered that is, we havev ≤ v 0 ≤ y 0 ≤ u 0 ≤û.