(p, q)-Equations with Negative Concave Terms

In this paper, we study a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction that has the combined effects of a negative concave term and of an asymmetric perturbation which is superlinear on the positive semiaxis and resonant in the negative one. We prove a multiplicity theorem for such problems obtaining three nontrivial solutions, all with sign information. Furthermore, under a local symmetry condition, we prove the existence of a whole sequence of sign-changing solutions converging to zero in C01(Ω¯)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1_0(\overline{\Omega })$$\end{document}.


Introduction
Let ⊆ R N be a bounded domain with a C 2 -boundary ∂ . In this paper, we study the following nonlinear Dirichlet problem Problem (1.1) is driven by the sum of two such operators with different exponents called the ( p, q)-Laplacian which is a nonhomogeneous operator. For such problems, we refer to the survey paper of Marano and Mosconi [13] and the references therein. In the right-hand side of (1.1), we have the combined effects of two distinct nonlinear terms. One term is the power function s → ϑ(x)|s| τ −2 s with 1 < τ < q and 0 > −c 0 ≥ ϑ(·) ∈ L ∞ ( ) which is a concave contribution (so (q − 1)-sublinear) to the reaction. The perturbation f : × R → R is a Carathéodory function, that is, s) is measurable for all s ∈ R and s → f (x, s) is continuous for a. a. x ∈ , which exhibits asymmetric growth as s → ±∞.
To be more precise, f (x, ·) is ( p − 1)-linear in the negative semiaxis (as s → −∞) and can be resonant with respect to the principal eigenvalue of (− p , W 1, p 0 ( )). In the positive semiaxis (as s → +∞), f (x, ·) is ( p −1)-superlinear but without satisfying the Ambrosetti-Rabinowitz condition (AR-condition for short). Hence, problem (1.1) is partly resonant and partly a concave-convex problem. In addition to this lack of symmetric behavior, another feature which distinguishes our work here from earlier ones on nonlinear elliptic equations with concave terms, is the fact that the coefficient ϑ : → R of the concave term is x-dependent and negative. In the past, problems with a negative concave term were studied by Perera [22], de Paiva and Massa [3], Papageorgiou et al. [20] for semilinear equations and by Papageorgiou and Winkert [15] for nonlinear equations driven by the ( p, 2)-Laplacian. From these works only the paper of Papageorgiou et al. [20] considers perturbations with asymmetric behavior as s → ±∞. In the literature, papers dealing with equations with concave terms assume that the coefficient is a positive constant. This is the case in the classical concave-convex problems, see Ambrosetti et al. [2] for equations driven by the Laplacian and by García Azorero et al. [5] for equations driven by the p-Laplacian. The difficulty that we encounter when we deal with equations that have negative concave terms is that the nonlinear strong maximum principle is not applicable, see Pucci and Serrin [23].

Preliminaries
In this section, we will recall the basic facts about the function spaces, the properties of the operator and some results of Morse theory.
To this end, let ⊆ R N be a bounded domain with a C 2 -boundary ∂ . For any r ∈ [1, ∞], we denote by L r ( ) = L r ( ; R) and L r ( ; R N ) the usual Lebesgue spaces with the norm · r . Moreover, the Sobolev space W 1,r 0 ( ) is equipped with the equivalent norm · = ∇ · r for 1 < r < ∞.
The Banach space is an ordered Banach space with positive cone This cone has a nonempty interior given by where n(·) stands for the outward unit normal on ∂ .
Let A r : W 1,r 0 ( ) → W −1,r ( ) = W 1,r 0 ( ) * with 1 r + 1 r = 1 be the nonlinear operator defined by where ·, · is the duality pairing between W 1,r 0 ( ) and its dual space W 1,r 0 ( ) * . This operator is bounded, continuous, strictly monotone, and of type (S + ), that is, Motreanu et al. [14,p. 40]. Let X be a Banach space, ϕ ∈ C 1 (X ) and c ∈ R. We introduce the following two sets is a topological pair such that Y 2 ⊆ Y 1 ⊂ X and k ∈ N 0 , then we denote by H k (Y 1 , Y 2 ) the k-th singular homology group for the pair (Y 1 , Y 2 ) with integer coefficients. If u ∈ K ϕ is isolated, the k-th critical group of ϕ at u is defined by The excision property of singular homology implies that the definition of C k (ϕ, u) is independent of the choice of the isolating neighborhood U , see Motreanu et al. [14]. The usage of critical groups allows us to distinguish between critical points of the energy functional.
We say that ϕ ∈ C 1 (X ) satisfies the Cerami condition (C-condition for short) if every sequence {u n } n∈N ⊆ X such that {ϕ(u n )} n∈N ⊆ R is bounded and (1 + u n X )ϕ (u n ) → 0 in X * has a strongly convergent subsequence. This is a compactness-type condition on the functional ϕ which compensates the fact that the ambient space X need not be locally compact.
For s ∈ R, we set s ± = max{±s, 0}. If u : → R is a measurable function, we Moreover, we denote by int Finally, the critical Sobolev exponent of p ∈ (1, ∞), denoted by p * , is given by

Multiple Solutions
In this section, we produce three nontrivial solutions of problem (1.1) where two of them have constant sign and one has changing sign. Now we introduce the hypotheses on the data of problem (1.1). H 1 : f : × R → R is a Carathéodory function such that f (x, 0) = 0 for a. a. x ∈ and it satisfies the following assumptions: (i) there exist r ∈ ( p, p * ) and 0 ≤ a(·) ∈ L ∞ ( ) such that for a. a. x ∈ and for all s ∈ R; uniformly for a. a. x ∈ and there exists uniformly for a. a. x ∈ ; (iii) there exist β 1 ∈ L ∞ ( ) and β 2 > 0 such that uniformly for a. a. x ∈ and for every λ > 0 there existsμ(λ) ∈ (1, β) such thatμ(λ) →μ ∈ (1, β) as λ → 0 + and f (x, s)s ≤ĉ λ|s|μ (λ) + |s| r −c|s| β for a. a. x ∈ , for all s ∈ R withĉ,c > 0.

Remark 3.2
Hypotheses H 1 (ii) and H 1 (iii) imply the asymmetric behavior of the perturbation f (x, ·). Indeed, hypothesis H 1 (ii) says that f (x, ·) is ( p − 1)-superlinear as s → +∞ but need not satisfy the AR-condition, see, for example, Ghoussoub [6, p. 59]. Our condition is less restrictive and allows also nonlinearities with "slower" growth as s → +∞ which fail to satisfy the AR-condition. Here, we refer to a unilateral version of the condition since it concerns only the positive semiaxis [0, ∞). Hypothesis H 1 (iii) says that f (x, ·) is ( p − 1)-linear as s → −∞ and can be resonant with respect to the principal eigenvalue of (− p , W 1, p 0 ( )). Note that in hypothesis H 1 (i), we want a ∈ L ∞ ( ) in order to be able to apply the regularity theory of Lieberman [12].

Example 3.3
The following function satisfies hypotheses H 1 but fails to satisfy the AR-condition: . Moreover, we introduce the positive and negative truncations of ϕ, namely, the C 1 -functionals ϕ ± : W Proof From hypotheses H 1 (iv), we see that for given ε > 0, we can find x ∈ and for all s ∈ R. (3.1) Using (3.1) and hypotheses H 0 , we get for u ∈ W Therefore, we find a number t 0 ∈ (0, ∞) such that Thus, ξ λ (t 0 ) = 0, and this implies Next, we show that ϕ + : W 1, p 0 ( ) → R satisfies the C-condition.

Proposition 3.5 Let hypotheses H 0 and H 1 be satisfied. Then the functional
Combining (3.2) and (3.5) yields for some c 4 > 0 and for all n ∈ N. Next, we take for all n ∈ N. Adding (3.6) and (3.7) and using hypotheses H 0 as well as τ < q < p, we get for some c 5 > 0 and for all n ∈ N.
Hypotheses H 1 (i) and H 1 (ii) imply that we can findβ 0 ∈ (0, β 0 ) and c 6 > 0 such thatβ for a. a. x ∈ and for all s ≥ 0. Using (3.9) in (3.8) leads to u + n μ μ ≤ c 7 for some c 7 > 0 and for all n ∈ N. Hence First, assume that p = N . From hypothesis H 1 (ii) it is clear that we may assume that μ < r < p * . Then we can find t ∈ (0, 1) such that Using the interpolation inequality (see Papageorgiou and Winkert [18,p. 116]), we have This combined with (3.10) results in for some c 10 > 0. Recall that p = N . If p > N , then by definition we have p * = ∞ and so If p < N , then we have by definition p * = N p N − p . So from (3.11) and H 1 (ii), it follows Hence, (3.14) holds again in this case.
Finally, let p = N . Then by the Sobolev embedding theorem, we know that W 1, p 0 ( ) → L s ( ) is continuous for all 1 ≤ s < ∞. Then, in the argument above, we need to replace p * by s > r > μ. We choose t ∈ (0, 1) such that Then, using (3.15), we have tr < p and so {u + Then there exists a subsequence, not relabeled, such that (3.4), pass to the limit as n → ∞ and use (3.16), we obtain By the monotonicity of A q , we have Using this in the limit above, we obtain Hence, from the convergence properties in (3.16), we conclude that Also, from Proposition 3.5, we know that Then, (3.17), (3.18), and (3.19) permit the usage of the mountain pass theorem. Therefore, we can find u 0 ∈ W 1, p 0 ( ) such that Hence, u 0 = 0. From Ho et al. [7,Theorem 3.1], we know that u 0 ∈ L ∞ ( ). Then the nonlinear regularity theory of Lieberman [12] implies that u 0 ∈ C 1 0 ( ) + \ {0}. Remark 3.7 Eventually, we will show that u 0 ∈ int C 1 0 ( ) + , see Corollary 3.12. However, at this point, due to the negative concave term, we cannot use the nonlinear Hopf maximum principle, see Pucci and Serrin [23, p. 120], and infer that u 0 ∈ int C 1 0 ( ) + . Next, we are looking for a negative solution of problem (1.1). So, we work with the functional ϕ − : W (3.20) From (3.20), we have  Note that from hypothesis H 1 (iii), we have  (3.22). From this as in the proof of Proposition 3.5, we conclude that ϕ − : W 1, p 0 ( ) → R satisfies the C-condition.
On account of hypothesis H 1 (iii), we see that (3.28) Then (3.28), Proposition 3.8, and the mountain pass theorem lead to the following result. In what follows S + (resp. S − ) denote the set of positive (resp. negative) solutions to (1.1). From Propositions 3.6 and 3.9, we have Next, we are going to prove that S + has a minimal element and S − a maximal one. So we have extremal constant sign solutions, that is, there is a smallest positive solution u * and a largest negative solution v * . These solutions will be useful in proving the existence of a sign-changing solution. Indeed, any nontrivial solution of problem (1.1) in the order interval [v * , u * ] distinct from v * and u * is necessarily sign-changing.
From (3.31), we have ψ + (u) = 0, that is, 0 ( ) in the equality above shows that u ≥ 0 with u = 0. Moreover, the nonlinear regularity theory of Lieberman [12] and the nonlinear strong maximum principle, see Pucci and Serrin [23, pp. 111 and 120], imply that u ∈ int C 1 0 ( ) + . Next, we show the uniqueness of this positive solution. For this purpose, we introduce the functional j : otherwise.
Let dom j = {u ∈ L 1 ( ) : j(u) < ∞} be the effective domain of j : Using the ideas of Díaz and Saá [4] along with the fact that the function s → sˆη τ for τ <η is increasing and convex, we know that j is convex. Let w ∈ W 1, p 0 ( ) be another positive solution of (3.30). As done before, we get w ∈ int C 1 0 ( ) + . From l'Hospital's rule, we have From (3.32), we know that w τ u τ ≤ c with c > 0 and so −w τ ≥ −cu τ . Then, for |t| small enough, we have Then the convexity of j implies that the directional derivative of j at u τ and at w τ , respectively, in the direction h exists. Moreover, using the nonlinear Green's identity, see Papageorgiou et al. [21, p. 35], we have The convexity of j implies the monotonicity of j . So, we have 0 ≤ĉ Proof Let u ∈ S + and consider the Carathéodory function l + : × R → R defined by We set L + (x, s) = s 0 l + (x, t) dt and consider the C 1 -functional σ + : W Moreover, it is also sequentially weakly lower semicontinuous. So, we can findũ ∈ W 1, p (3.34) Since τ < q < p < r , we see that σ + (ũ) < 0 = σ + (0). Hence,ũ = 0. From (3.34), we have σ + (ũ) = 0. This gives This yields by applying (3.33) along with (3.29) and the fact that u ∈ S + Hence,ũ ≤ u. So we have proved that From (3.36), (3.33), and (3.35), it follows thatũ is a positive solution of (3.30). Theñ u = u ∈ int C 1 0 ( ) + and so u ≤ u for all u ∈ S + . Similarly, we show that v ≤ v for all v ∈ S − .
We have the following corollary.

Corollary 3.12 Let hypotheses H 0 and H 1 be satisfied. Then
Now we are ready to produce extremal constant sign solutions.

Proposition 3.13
Let hypotheses H 0 and H 1 be satisfied. Then there exist solutions u * ∈ S + and v * ∈ S − such that Proof From Papageorgiou et al. [19,Proposition 7], we know that S + is downward directed. So, using Lemma 3.10 of Hu and Papageorgiou [8], we can find a decreasing sequence {u n } n∈N such that inf n∈N u n = inf S + .
Since u n ∈ S + , we have (3.38) Choosing h = u n − u in (3.37), passing to the limit as n → ∞, and using the convergence properties in (3.38), we obtain lim sup n→∞ A p (u n ), u n − u ≤ 0.
Then, by the (S + )-property of A p , we get Passing to the limit in (3.37) and using (3.39), we have From Proposition 3.11, we know that u ≤ u * . Hence, u * ∈ S + and u * ≤ u for all u ∈ S + .
Similarly, we produce v * ∈ S − such that v ≤ v * for all v ∈ S − . Note that S − is upward directed.
Using the extremal constant sign solutions obtained in Proposition 3.13, we are going to prove the existence of a sign-changing solution. As explained earlier, we focus on the order interval [v * , u * ] and look for solutions in [v * , u * ] \ {0, u * , v * }. Such a solution turns out to be sign-changing. Implementing the approach just described, let u * ∈ S + and v * ∈ S − be the extremal constant sign solutions from Proposition 3.13 and consider the truncation functions k 1 , k 2 : × R → R defined by and We set and consider the C 1 -functionals ζ, ζ ± : W Applying (3.40), (3.41), (3.42), and (3.43), we check easily that Due to the extremality of u * and v * , we conclude that (3.46) since τ < q < p, for t ∈ (0, 1) small enough, we have by using H 1 (iv) and choosing Due to (3.46), we know thatũ * ∈ K ζ + and soũ * = u * , see (3.45). Let > 0 and Since ζ | C 1 0 ( ) + = ζ + | C 1 0 ( ) + , we obtain for u ∈ B (3.47) We write as abbreviation Then, for the first integral on the right-hand side in (3.47), we have (3.48) From H 1 (iv), for given ε > 0, we can findĉ 11 =ĉ 11 (ε) > 0 such that for a. a. x ∈ and for all s ∈ R. Using (3.49), the second integral on the right-hand side in (3.47) can be estimated by (see also the proof of Proposition 3.4)  Recall that u * ∈ C 1 0 ( ) + \ {0} and u ∈ B Thus, |{−u − ≤ v * }| N → 0 as → 0 + and |{v * ≤ −u − }| N > 0 for > 0 small enough and it is also decreasing in . Then, for λ small and for > 0 small enough, from (3.51), it follows that u * is a local C 1 0 ( )-minimizer of ζ and from Papageorgiou and Rȃdulescu [16], we deduce that u * is a local W 1, p 0 ( )-minimizer of ζ . Similarly, working with ζ − instead of ζ + , we can show the result for v * ∈ S − . Now we are ready to generate a sign-changing solution for problem (1.1).

Proposition 3.15
Let hypotheses H 0 and H 1 be satisfied. Then problem (1.1) has a sign-changing solution y 0 ∈ [v * , u * ] ∩ C 1 0 ( ). Proof We assume that K ζ is finite, otherwise on account of (3.45), (3.40), and (3.41), we would have infinity smooth sign-changing solutions. Moreover, we assume that ζ(v * ) ≤ ζ(u * ). The analysis is similar if the opposite inequality holds. From Proposition 3.14, we know that u * is a local minimizer of ζ . Recall that the functional ζ is coercive. So, it satisfies the C-condition, see, for example, Papageorgiou et al. [21, p. 369]. So, using Theorem 5.7.6 of Papageorgiou et al. [21], we can find ρ ∈ (0, 1) small enough such that Therefore, we can use the mountain pass theorem and find y 0 ∈ W 1, p 0 ( ) such that From (3.53), we see that y 0 / ∈ {v * , u * }. Moreover, Theorem 6.5.8 of Papageorgiou et al. [21] implies that On the other hand, the presence of the concave term and the C 1 -continuity of critical groups imply that  (3.44), where the difference is that, due to the local oddness of f (x, ·), we truncate in (3.40), (3.41) above at int C 1 0 ( ) + η < min{γ, u * } instead of u * and below at − int C 1 0 ( ) + −η > max{−γ, v * } instead of v * .
Since V is finite dimensional, all norms are equivalent. Therefore, we can find ρ V > 0 such that u ∈ V and u ≤ ρ V imply |u(x)| ≤ δ for a. a. x ∈ . (4.2) Applying (4.1) and (4.2), we have for u ≤ ρ V see the truncations in (3.40) and (3.41). Recalling that η > ϑ ∞ , we choose ε ∈ (0, η − ϑ ∞ ). Then, using once more the fact that on V all norms are equivalent, we obtain  On account of Proposition 4.1, we can apply Theorem 1 of Kajikiya [9] and obtain a sequence {u n } n∈N ⊆ W 1, p 0 ( ) such that u n ∈ K ζ for all n ∈ N and u n → 0 in W 1, p 0 ( ).

Remark 4.3
It will be interesting to extend the results of this paper to anisotropic equations. We believe that this is feasible. However, concerning possible extensions to double-phase problems with unbalanced growth, we doubt that this is possible due to the lack of a global regularity theory for such problems.