Existence and regularity results for nonlinear elliptic equations in Orlicz spaces

We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on $(x,u,\nabla u)$, and with a convective term $f$. The assumptions on the members of the equation are formulated in terms of Young's functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.


Introduction
In the present paper we deal with the existence and regularity of solutions to the the following nonlinear elliptic problem (1.1) −div(A(x, u, ∇u)) = f (x, u, ∇u) in Ω u = 0 on ∂Ω .
Ω is a subset of R n having finite measure, A : Ω × R × R n → R n and f : Ω × R × R n → R are Carathéodory functions.
In spite of the many boundedness (see, for instance, [BaCiMa,Ci2,LU,MaWi] and the references therein) or regularity (see [BaCoMi,BPS,DBe,Li,Li1,Mi]) results, in which the existence of a solution is assumed a priori, there are fewer results about the existence of solutions for (1.1), unless the principal part is the p−Laplacian or growths like the p−Laplacian.This is due to the fact that when the operator A in (1.1) depends also from the unknown u, technical difficulties arise.First of all, variational methods are not directly applicable (and this holds also for problems with a convective term, namely with a nonlinearity f depending also on ∇u), and the properties of A are actually well known only in standard contexts (see [CLM, Tr]).This means, in The author is member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni particular, that in most situations, the abstract framework for the study of (1.1) is the classical Sobolev space W 1,p 0 (Ω).In this regard, we note that there are many papers dealing with the existence of solutions to (1.1), whenever the principal part does not depends explicitely on u, and the tecniques used are the most diverse.This also testifies to the growing interest in recent years for the study of these problems.The two main motivations of the paper are to investigate problems whose differential part depends explicitly also on the unknown, and to remove restrictions of power type growth in the study of (1.1).To do this, we have to establish some properties of A, which are of independent interest (see Proposition 2.17).Just to give an idea, in our contest we can manage problems driven by an operator having a power-times-logarithmic type growth, like (see Example 2.16) To the best of our knowledge no results are actually available for problems driven by such operators.Conversely, many existence and regularity results are available for (1.1), when A is the p-Laplacian, or the (p, q)-Laplacian, namely when A(∇u) = |∇u| p−2 ∇u or A(∇u) = |∇u| p−2 ∇u + |∇u| q−2 ∇u (see [AAP, AF, BF, FMP, LMZ, MW, NS, R, Z]).There is also an extensive literature concerning problems in which the structure of A allows to tackle (1.1) with the same techniques adopted for the p−Laplacian.We refer to [FM, PZ] for problems with A(∇u), to [Boc,Boc1,BNV,HT,MW,NS,T] for problems with A(x, ∇u) and finally, for problems with A(x, u, ∇u) in W 1,p 0 (Ω), we cite [FMMT, G].Much less is available regarding more general operators, built via a function radial with respect to ξ, but not necesseraly a polinomyal.This new situation requires the use of Young functions and Orlicz spaces.Existence (and regularity) results for problems with A(∇u) = a(|∇u|)∇u can be found in [BaTo1,BaTo2,CGSS,FMST].In [DF] the authors deal with an operator depending on the three variables via Young functions of a real variable.In this paper, the growth conditions on the terms appearing in (1.1) require to replace the customary Sobolev space with an Orlicz space (see (1.2)).These conditions cover several instances already studied in the papers cited above.We stress that Young's functions are also involved in the growth of the convective term f .Similar hypotheses can be found in [BaTo1,BaTo2,DF].Given the non-variational nature of the problem, we use the method of sub and super solutions, togheter with truncation techniques and the theorem of existence of zeros for monotone operators.For the regularity results, a main tool is a Theorem due to Lieberman (see [Li1,Theorem 1.7] and Proposition 4.1).The first step necessary to obtain our results consists in establishing some properties of the operator A in Orlicz spaces.This properties, as well as basic definitions and some other auxiliary results, are collected in Section 2. The main existence theorems (and a general example of function A satisfying our assumptions) are presented in Section 3: beside to a general existence result, Theorem 3.4, in Theorem 3.8 we consider a special instance of (1.1), where a subsolution and a supersolution are obtained via variational methods.In Section 4 we prove our regularity results.Theorems 3.8 and 4.4 have among the main hypothesis, the existence of a subsolution and a supersolution.Starting from them, we construct a suitable functional, in which appropriate truncations of A and of the convective term f are involved.Finally, Section 5 is devoted to some examples where we highlight how the method of sub and super solutions works well in all situations where the convective term f has two zeros of opposite sign, namely f (x, s 1 , 0) = f (x, s 2 , 0) = 0 for all x ∈ Ω and for some s 1 , s 2 ∈ R, enjoying the condition In this regard, it should be noted that the use of the method of sub and super solutions, combined with truncation techniques, is extremely valid when a mix of conditions related to both A and f It is not restrictive to assume that any finite valued Young function is continuous.For Young functions (2.1) A(λt) ≤ λA(t) for λ ≤ 1 and t ≥ 0.

Definition 2.2
The Young conjugate of a Young function A is the Young function A defined as The following inequalities are a consequence of Definition 2.2 (2.2) The inverse function in (2.2) is the generalized right continuous inverse.In general a Young function is left continuous (in effect it is continuous, unless it takes the value +∞), and when we deal with the inverse we consider the right continuous one.
Definition 2.3 A Young function A is said to dominate another Young function B near infinity, if there exist constants k > 0 and M ≥ 0 such that If (2.4) holds with M = 0, then we say that A dominates B Two Young functions A and B are called equivalent near infinity (globally) if they dominate each other near infinity (globally) and w e briefly write A ≈ B near infinity (globally).
Definition 2.4 Two functions f, g : (0, ∞) → [0, ∞), are equivalent near infinity (briefly f ≈ g near infinity) if and only if there exist suitable positive constants c 1 , c 2 and s 0 such that Definition 2.5 A Young function B is said to increase essentially more slowly than A near infinity (briefly B ≪ A), if B is finite valued and where A −1 denotes the right continuous inverse of A. Note that if A(t) = +∞ for t > t 0 , then A −1 (s) is constant for s sufficiently large, whatever inverse we take.
Proposition 2.6 Let A and B be two Young functions, such that B ≪ A. Then there exists a Young function One can easily check that C −1 is concave, strictly increasing (inasmuch A and B are strictly increasing), and Definition 2.7 A Young function A is said to satisfy the ∆ 2 -condition near infinity (briefly A ∈ ∆ 2 near infinity) if it is finite valued and there exist two constants K ≥ 2 and M ≥ 0 such that (2.9) A(2t) ≤ KA(t) for t ≥ M .
We give basic definitions and the main properties on the Orlicz spaces.Let Ω be a measurable set in R n , with n ≥ 1.Given a Young function A, the Orlicz space L A (Ω) is the set of all measurable functions u : Ω → R such that the Luxemburg norm , and it is a Banach space (see [Ad]).If A and B are Young functions, A ≈ B near infinity and |Ω| < ∞ then L A (Ω) = L B (Ω), and there exist If A is a Young function and A denotes its conjugate, then a generalized Hölder inequality (2.12) holds for every u ∈ L A (Ω) and v ∈ L A (Ω).If A ∈ ∆ 2 globally (or A ∈ ∆ 2 near infinity and Ω has finite measure) then (2.13) Also, if B ≪ A near infinity and Ω has finite measure, then are equivalent to the fact that A ∈ ∇ 2 ∩ ∆ 2 globally (near infinity).The following result extends Lemma A.1 of [CM] to Young functions A ∈ ∇ 2 near infinity.
Lemma 2.9 Let Ω be a subset of R n , with finite measure.Let A be a Young function.Assume that i ∞ A > 1.Then there exists k 3 (i ∞ A , |Ω|) ≥ 0 such that Here k 1 is that of (2.11).

Proof. Since i
A and consider the Young function where c 1 and c 2 are chosen in such a way that The Lemma below is a consequence of the Theorems of Vitali and De la Vallée-Poussin.It will be used several times in the paper.
Lemma 2.10 Let Ω ⊆ R n be such that |Ω| < ∞ and let A be a Young function.
Proof.Let B ≪ A near infinity.Then B is finite valued and we may assume, without loss of generality, that B is continuous.We must prove that lim k→+∞ Ω B |u k −u| λ dx = 0, for all λ > 0.
If A(t) ≡ +∞ for t > t 0 > 0, then L A (Ω) = L ∞ (Ω) and the conclusion follows via the dominated convergence theorem.If A is finite valued, then we may assume that it is continuous.Fix λ > 0 and let M > 0 be such that u k L A (Ω) ≤ M for all n ∈ N .Then, the continuity of A and the Fatou's lemma guarantee that u ∈ L A (Ω) and is equintegrable and we can apply the Vitali's Theorem.This For the last part of the proof we observe that |u k − u| ⇀ v in L A (Ω), up to a subsequence.From the proof above we know that |u k − u| → 0 in L B (Ω).Thus v ≡ 0. This apply to all the subsequences, thus it holds for the whole sequence.
From now and throughout the paper we assume that Ω is an open set in R n with |Ω| < ∞.The isotropic Orlicz-Sobolev spaces W 1,A (Ω) and W 1,A 0 (Ω) are defined as The spaces W 1,A (Ω) and W 1,A 0 (Ω) equipped with the norms (2.22) are Banach spaces.The norm on W 1,A 0 (Ω) is equivalent to the standard one Definition 2.11 The optimal Sobolev conjugate of A is defined by where H : [0, ∞) → [0, ∞) is given by provided that the integral is convergent.Here, H −1 denotes the generalized left-continuous inverse of H.
Example 2.12 A general Young function Thus, given any Young function A such that A(t) ≈ t p lg q (1 + t) near infinity , (2.33) and a set Ω with finite measure, it holds W 1,A 0 (Ω) = W 1,B 0 (Ω), up to equivalent norms.The conjugate A and the optimal Sobolev conjugate of A satisfy near infinity in all the other cases . (2.35) All the examples of the paper will involve functions A complying with (2.33).
We recall now some definitions and the theorem on pseudomonotone operators.Then we introduce the conditions on the function A in (1.1).Definition 2.13 Let X be a real reflexive Banach space.A mapping B : X → X * is called (ii) bounded if it maps bounded sets into bounded sets; Theorem 2.14 (see [CLM,Theorem 2.99]) Let X be a real reflexive Banach space and let B : X → X * be a bounded, coercive and pseudomonotone operator.Then, for every b ∈ X * the equation Bx = b has at least solution x ∈ X.
Let us now introduce the hypotheses on the function A operator and verify the properties of the integral operator associated with it, in a standard way.We note that this is a Leray Lions type operator.The properties of the auxiliary truncated operator will easily follow from those of the non-truncated operator (see Proposition 2.17).We extend the results in [CLM, Tr] in several directions: Orlicz spaces are considered, and even in the case of Lebesgue spaces, the hypotheses are slightly more general.
Let Ω ⊂ R n be a set with finite measure and let A be a Young function, A ∈ ∆ 2 ∩ ∇ 2 near infinity.

Consider the vector valued function
, enjoying with the properties that each a i (x, s, ξ) is a Carathéodory function, and Remark 2.15 Using the condition A ∈ ∆ 2 near infinity, and |Ω| < ∞, it can be shown that (2.38) is equivalent to We will use (2.39) in the sequel.
Finally, when p = n, q = n − 1 we consider and then with Let A(t) be defined as in (2.33).Taking into account the conditions in (2.41) and in (2.42), for the function a in (2.47) and (2.48) it holds a(x) When (2.42) holds we choose the Young function F (s) ≈ e s β near infinity.Similarly, when (2.43) is in effect, then we choose F (s) ≈ e e s β near infinity.In all the three cases F ≪ A n near infinity and A −1 (F (|s|)) is equivalent, near infinity, to the function in the s variable, appearing in (2.47), (2.48) and (2.49) respectively.Then (2.36) holds with q(x) = a(x) + k, for some k > 0.
We now consider the borderline instances concernig β.When β = 0, then we can take r = p δ .When β = nδ n−p in (2.41), then in addition to and a ∈ L ∞ (Ω).When q(x) ≡ 0 then we have the classical p-Laplacian.

Define now the operator
The properties of S are listed in the next proposition.
We can thus apply the Lebesgue theorem to obtain The arguments above apply to any subsequence of {u k }.This means that given any subsequence, we can extract a subsubsequence for which (2.54) holds.Thus (2.54) holds for the whole sequence.Note that We divide the proof in four steps.
Let F be as in (2.36).From Proposition 2.6 there exists a Young function F 1 such that F ≪ F 1 ≪ A n near infinity.Thus {u k } strongly converges to u in L F 1 (Ω), and we can find a function g ∈ L F 1 (Ω) and a subsequence of The function v λ ∈ L 1 (Ω) because of (2.14) and standard arguments used several times in this proof.Thus, arguing as for (2.54), A(x, u k , ∇u) − A(x, u, ∇u) → 0 in L A (Ω) and (2.58) By definition of weak convergence (2.59) From (2.37) Passing to the limit in (2.60) and using (2.56), (2.59) and (2.58) , and we can find a subsequence, say still {u k }, and a set Ω 0 ⊂ Ω, such that |Ω 0 | = 0 and (recall that We prove that for every x ∈ Ω \ Ω 0 there exists M > 0 such that |∇u k (x)| ≤ M for all k ∈ N .We argue by contradiction.Assume that there exists x ∈ Ω \ Ω 0 such that for every h > |∇u(x)| + 1 there exists k h ∈ N such that |∇u k h (x)| > h.In particular The sequence converges to ξ ∈ R n , up to a subsequence.We keep the same notation as above for the subsequence and use (2.37) and (4.14) From (2.64) and (2.62) and this leads to (A(x, u(x), ∇u(x) + ξ) − A(x, u(x), ∇u(x))) ξ = 0, namely ξ = 0.This contradicts |ξ| = 1, thus |∇u k h (x)| ≤ M , for some M > 0 and we can find a subsequence, say still {∇u k h }, converging to η ∈ R n .Then from (2.62) and the convergence of From (2.37) we deduce η = ∇u(x).We have so proved that every subsequence of {∇u k } has a subsequence converging to ∇u(x).Thus the whole sequence converges to ∇u(x) in Ω \ Ω 0 . Step Since A ∈ ∇ 2 near infinity we can apply Corollary 2.10 to obtain So, using (2.68), (2.56) and Step 2 (2.69) Step 4 Step 3 and the assumption G ≪ A n , we know that there exists a subsequence of {u k }, say still {u k } and a function g in Ω, we deduce that (2.70) A standard and repeatedly used argument shows that it holds for the whole sequence.Equation (2.70) and the ∆ 2 condition on A, guarantees that ∇u k → ∇u in L A (Ω).
We now construct the truncation of S that we use in the proof of our results.For every r ∈ R, we set r + = max{r, 0}, r − = max{−r, 0}.Let u, u ∈ W 1,A (Ω) be such that (u) − , u + ∈ W 1,A 0 (Ω), and u ≤ u a.e. in Ω.The truncation operator The properties of u, u guarantee that T (u), (u − u) + , (u − u) − ∈ W 1,A 0 (Ω).In particular T is well defined.It is known (see [H], p.20) that (2.72) a.e. on the set {u > u} ∇u(x) a.e. on the set {u ≤ u ≤ u} ∇u(x) a.e. on the set {u < u} Given the functions u, u ∈ W 1,A (Ω) as above, and such that u, u ∈ L An (Ω), let us define the operator S T : Proof.The inequality |T (u(x))| ≤ |u(x)| for all x ∈ Ω guarantees that the operator S T is well defined and bounded.Thanks to Lemma 4.1 in [BaTo1], the arguments used in Proposition 2.17 work also for S T .Thus S T is continuous and has the (S) + property.

Main results
In this Section we state two of the main results of the paper (Theorems 3.4 and 3.8).
First we give the fundamental definitions of weak solution, subsolution and supersolution to (1.1). and Then problem (P ) has a solution u ∈ W 1,A 0 (Ω) such that u ≤ u ≤ u a.e. in Ω.
To prove Theorem 3.4, we perturb problem (1.1) and formulate an auxiliary one.Let Π : Let N f • T : W 1,A 0 (Ω) → (W 1,A 0 (Ω)) * be the operator defined as Given µ > 0, we consider the following problem The result below guarantees that problem (3.3) has a solution, provided the parameter µ > 0 is sufficiently large.
Remark 3.6 The proof above works also if we weaken (2.36). ..(2.38), requiring them to hold for s ∈ [u(x), u(x)] rather than for all s ∈ R.This is because in the proof we consider only the truncated function A(x, T u, ∇u).
Prooof of Theorem 3.4.By Theorem 3.5 there exists a solution u ∈ W 1,A 0 (Ω) of the truncated auxiliary problem (3.3) provided µ > 0 is sufficiently large.Let us fix such a µ and u.Via the same comparison arguments of the proof of Theorem 3.6 of [BaTo1] we can prove that the solution of (3.3) has the enclosure property u ∈ [u, u].Thus, it follows from (2.71) and (3.2) that T u = u and Π(u) = 0. Consequently, u is a solution of(1.1).
The proof of the Corollary below follows the same lines as that of Corollary 5.2 of [BaTo1].
We consider now a special instance of (1.1), in which A does not depend on s and has a potential with respect to ξ.So, let Ω ⊂ R n be a set of finite measure and let A, B be two Young functions such that A ∈ ∆ 2 ∩ ∇ 2 near infinity, B ∈ ∇ 2 near zero, and A • B −1 is a Young function too.We assume that A : Ω × R n → R n , A = (a 1 , . . .a n ), is such that each a i (x, ξ) is a Carathéodory function, and Furthermore, we assume that there exists a measurable function Φ : Ω × R n → R, even with respect to ξ ∈ R n and such that Since A • B −1 is a Young function, it follows that A dominates B near infinity and B dominates A near zero.Thus Condition (3.10) ensures that Φ(x, •) is convex for every x ∈ Ω.From (3.9) and (3.11) there exist k 4 , k 5 > 0 such that For functions f satisfying suitable growth conditions we can construct a sub or a supersolution for problem (3.15), via variational methods.The ∆ 2 and ∇ 2 conditions play a crucial role here.
Proof.Suppose that (3.16) is in force.We construct a subsolution u ≤ 0 a.e., u ≡ 0, and show that u ≡ 0 is a supersolution but not a solution to (3.15).Then, we show that f satisfies (3.8).Put G 1 (t) = t 0 g 1 (τ )dτ, t ≥ 0 and consider the functional J : W 1,A 0 (Ω) → R, defined as We prove that J is well defined, weakly lower semicontinuous, coercive and for all u, v ∈ W 1,A 0 (Ω).We examine separately the three integrals.Due to (3.14), the fact that A ∈ ∆ 2 at infinity, and the convexity of Φ(x, •), for all x ∈ Ω, the functional u → Ω Φ(x, ∇u)dx is well defined in W 1,A 0 (Ω), convex.We briefly sketch the proof of its regularity, because it makes use of standard arguments like the Lebesgue Theorem, and the properties of Young's functions.Let u, v ∈ W 1,A 0 (Ω).For all x ∈ Ω, all t > 0, t << 1, there exists µ t,x ∈ (0, 1) such that We used (3.9), the monotonicity of A −1 • A and B −1 • B, and (2.2).Now, the condition A ∈ ∆ 2 near infinity, (3.13) and the Lebesgue Theorem, allow to prove that the functional u → Ω Φ(x, ∇u)dx is C 1 .Thus the weak lower semicontinuity of J and equation (3.19) follow.
This proves that J is coercive.Thus it has a global minimum.Let u be a global minimum point for J.We prove that u ≡ 0. To this end consider a function v ∈ C 1 0 (Ω), such that b|∇v(x)| ≤ t and k 5 k|∇v(x)| ≤ t for all x ∈ Ω.Also, v ≤ 0 and ρ 1 (x)v(x) ≡ 0 in Ω.The inequality B(t 1 ) B(t 0 ) > t 1 t 0 k B holds for 0 < t 0 < t 1 < t, and some k B > 1, by virtue of the ∇ 2 condition near zero.Then, choosing once t 1 = b|∇v|, t 0 = bt|∇v|, and secondly t 1 = k 5 k|∇v|, t 0 = tt 1 , with t < 1, and taking into account (3.14) and this proves that J(u) < 0. Using J(−|u|) ≥ J(u) and the fact that Φ(x, •) is even, we obtain u ≤ 0 a.e. in Ω.Now we prove that u is a subsolution and u ≡ 0 is a supersolution but not a solution to (1.1).Note that Acting with any v ∈ W 1,A 0 (Ω), v ≥ 0, in (3.22) and using (3.16) that is u is a subsolution to (3.15).Using (3.16) and choosing thus u ≡ 0 is a supersolution to (1.1) and the assumption on f (x, 0, 0) guarantees that it is not a solution.

Regularity results
In this section we give some existence and regularity results, Theorems 4.4 and 4.5.We strenghten the hypotheses on Ω and on A (see (4.1)), in order to apply regularity theory (see Proposition 4.1).The proof of the existence is based on sub and supersolution methods, while the main tool for the regularity is Theorem 1.7 of [Li1] (see also the remark after that result and [Li]), that we recall below.
Let A : Ω×R×R n → R n be a vector valued function, with Carathéodory components, a i , i = 1, . . ., n.
Suppose A and f satisfy the structure conditions (here a ij (x, s, η) for some positive constants Λ, Λ 1 , M 0 , for all x and y ∈ Ω, for all s, w ∈ [−M 0 , M 0 ] and for all ξ ∈ R n .Then, any solution u ∈ W 1,A (Ω), with |u| ≤ M 0 in Ω, is C 1,β (Ω) for some positive β.
Following the spirit of Example 5.1, we present a function A satisfying (4.2), (4.3) and (4.4).This general function will be used in Example 5.3, but we prefere to introduce this function here, to emphasize its general structure.
Here Ω is a bounded domain with a C 1,α boundary, β, δ ≥ α and p, q are like above.We show that A satisfies (4.2), (4.3) and (4.4).It holds This guarantees that n i,j=1 and (4.2) holds.A simple calculation shows that (4.3) holds too.For what concerns (4.4), let M > 0 and take x, y ∈ Ω, s, w For the last inequality in (4.12) we used (4.15) and (4.16) further down.They are obtained via the inequalities below.If ρ ≤ 1 then there exists c > 0 such that The same holds for | x γ − y γ |, x, y ∈ Ω.Thus (2.39) remains true.
For the first existence and regularity Theorem we assume that problem (1.1) admits a subsolution and a supersolution u, u ∈ W 1,∞ (Ω).
Theorem 4.4 Let Ω be a bounded domain in R n with a C 1,α boundary.Let the functions A and A be as in Proposition 4.1.Assume further that (4.2), (4.3) and (4.4) hold for all s ∈ R. Let u, u ∈ W 1,∞ (Ω) be a subsolution and a supersolution for problem (1.1), with u(x) < u(x) a.e.x ∈ Ω.
The function g : R → R is continuous, g(s) > 0 for s ∈ [0, s) and g(s) = 0. First of all we note that u 1 = 0 and u 2 = s are a subsolution and a supersolution to (5.1) and u ≡ 0 is not a solution.The growth of the functions h and k quarantee that f satisfies (3.1).Due to the continuity of b, and taking into account Example 2.16 (with β = β 1 = 0), we see that conditions (2.36), (2.37) and (2.38) hold, for a.e.x ∈ Ω, all s ∈ [0, s], all ξ ∈ R n .Thus, by Theorem 3.4, problem (5.1) has a nontrivial solution u ∈ [u 1 , u 2 ].The same arguments work for different choices of h and g.In particular we see that Theorem 3.4 works well with all nonlinearities having two zeros s 1 and s 2 , with s 1 < 0 < s 2 , (f (x, s 1 , 0) = f (x, s 2 , 0) = 0 for all x ∈ Ω) and f (x, s, 0) has constant sign for s ∈]s 1 , s 2 [, or for which f (x, 0, 0) has constant sign and there exists s ∈ R such that f (x, s, 0) ≡ 0 and s • f (x, 0, 0) > 0.
(GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).This research is partially supported by the Ministry of Education, University and Research of Italy, Prin 2017 Nonlinear Differential Problems via Variational, Topological and Set-valued Methods (Project No. 2017AYM8XW).
continuity af S. Let us now demonstrate the (S) + property.Let {u k } be a sequence in W 1,A 0 (Ω), u k ⇀ u and (2.56) lim sup k→+∞ Su k , u k − u ≤ 0 .