Cent. Eur. J. Math. • 12(1) • 2014 • 114-127 DOI: 10.2478/s11533-013-0329-2

VERS ITA

Central European Journal of Mathematics

Generalized n-Laplacian: boundedness of weak solutions to the Dirichlet problem with nonlinearity in the critical growth range

Research Article

Robert Cerny1*

1 Department of Mathematical Analysis, Charles University, Sokolovska 83, 186 00 Prague 8, Czech Republic

Received 15 November SOtS; accepted SB February S013

Abstract: Let n > 2 and let 0 c Rn be an open set. We prove the boundedness of weak solutions to the problem u e W01 L*(0) and - divj$'(|Vu|) ^J + V(x)4>'(|u|) U = f(x,u) + ph(x) in 0,

where $ is a Young function such that the space W01L*(O) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h e (L*(0))* is a non-trivial continuous function and p > 0 is a small parameter. We consider two classical cases: the case of 0 being an open bounded set and the case of 0 = Rn.

MSG: 46E35, 46E30, 26D10

Keywords: Orlicz-Sobolev spaces • Trudinger embedding • Moser-Trudinger inequality • Moser iteration © Versita Sp. z o.o.

1. Introduction

In this note, we combine the methods from [7] and [16] with the generalized Moser-Trudlnger Inequality to prove the boundedness of weak solutions to the problem

u G W>nfi) and - div(0'(|Vu|) i^j + V(x)V(\u\) u = f(x,u)+ ^h(x) In Q, (1)

E-mail: rcerny@karlin.mff.cuni.cz

Springer

where 0 Is a Young function (0: [0, oo) i—> [0, oo) Is Increasing, convex, 0(0) = 0 and 0(t)/t ^ to, t ^ to) such that the space W0L0(Q) is embedded into an exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V(x) is a continuous potential, h e (L0(Q))* is a nontrivial continuous function, j > 0 is a small parameter and Q is either an open bounded set or Q = Rn. The precise assumptions on 0(t), f(x, t) and V(x) are given below (let us already note that these assumptions are different for Q bounded and Q = Rn).

The existence of nontrivial weak solutions to (1) was obtained in the paper [1] (see also [2-4]). The boundedness of weak solutions to (1) has two applications. First, this property of solutions can simplify several proofs in papers [1-4]. The second application follows from the fact that if Q is bounded and h(x) is a bounded function, then each bounded solution u(x) to (1) also solves the problem

u e WL0(Q) and - dl^0'(|Vu|) j = g(x) in Q, (2)

g(x) = f (x,u(x)) + Jh(x) - V(x)0'(|u|) u

is a bounded function (by our assumptions on f and V given below). For bounded weak solutions to (2), one has the results from [18] concerning the upper bound for the LTO-norm, the Holder continuity, the Harnack inequality and the Holder continuity of the gradient (notice that even though the results in [18] are stated for classical solutions, the proofs deal with the definition of the weak solution only). Similarly, if Q is not bounded, one can try to apply the results from [18] to bounded subsets of Q (in the case of unbounded Q, V(x) is not usually bounded and thus we do not have the uniform LTO-estimate of g(x) in (2)).

In the rest of this section we recall the generalized Moser-Trudinger inequality (a version of the famous Moser's result [19]), the existence results concerning equation (1) and we state our main theorem concerning the boundedness of weak solutions to (1).

Generalized Trudinger embedding and generalized Moser-Trudinger inequality

First we recall the classical Trudinger embedding and the Moser-Trudinger inequality. Let Q C Rn be an open bounded set. If W01,p(Q) denotes the usual completition of Qf (Q) in W1p(Q), then the Sobolev Embedding Theorem states that

W01,p(Q) C Lnp/(n—p)(Q) if 1 < p < n, and W01,p(Q) C (Q) if n < p.

In the borderline case p = n we have from the above

W0l,n(Q) C Lq(Q) for every q e [1,oo), but W01,n(Q) C Lto(Q). (3)

The lack of an optimal target space for the Sobolev em bedding of W0 (Q) among Lebesgue spaces inspired Trudinger [ ] to consider Orlicz spaces as the target spaces. He showed that

W01,n(Q) C L0(Q),

where L0(Q) is the Orlicz space corresponding to the Young function 0(t) = exp(tn/(n-1)) — 1 (the same results were independently obtained by Yudovich [23] and Pokhozhaev [21]). Behavior of the Trudinger embedding is further described by Moser's inequality [19]

ueW01,n(Q) IVu||Ln(Q)<1

f exp(K|u(x)|n/(n—1)) dx

< C(n, K, |Q|), K < nwj

1/(n—1)

K > nail——

where 0)n—1 denotes the surface of the unit sphere and |0| Is the n-dlmenslonal Lebesgue measure of 0. We also use this notation in the rest of the paper. Now, we proceed to some generalizations.

First, we start with the notation. By L0(0) we denote an Orlicz space corresponding to a Young function 0. This space is equipped with the Luxemburg norm

Mli*(fi) = lnf

A > 0 : /01 M I dx < 1

We define an Orllcz-Sobolev space WL0(Q) as the set WL0(Q) = {u : u, |Vu| G L*(Q)} equipped with the norm

||u||WL»(Q) = ||u||L»(Q) + ||Vu 1 L'(O),

where Vu is the gradient of u and we use its Euclidean norm in Rn. We put W0L*(Q) for the closure of Cq° (Q) in WL*(Q). If Q is bounded, on W0L*(Q) we can also use the Dirichlet norm ||u||w0l*(Q) = I|Vu||l*(q). which is equivalent to the standard Sobolev-type norm given above.

For f G N and a < n — 1 let us define the following constants (related to the Moser-type inequality):

fBVBni,>Yln f = 1

n a n „ ,, "Wn—1, f ~ ',

Y = -1- > 0, B =1--- = --— > 0, Kfna = i

" — 1 — a n — 1 (n — [B1lBMyJl1, f > 2.

To simplify our notation, for j G N, we set

log^] t = log t and log^jj t = log 1og|j—1] t for j > 2, exp[1] t = exp t and exPj] t = exp exp|j— 1] t for j > 2.

The space W0L'(Q) with

llm 0(t)a = 1, a < n — 1, t^j tn loga t

is embedded into an Orlicz space with the Young function that behaves like exp tY for large t. Moreover in the limiting case a = n — 1 we have an embedding into a double exponential space, i.e. the space WoL0(0) with

llm —--= 1, a <n — 1,

tn logn 11 log(2] t

is continuously embedded into a double exponential Orlicz space with the Young function that behaves like exp[2] tY for large t. Furthermore, in the limiting case a = n — 1 we have an embedding into a triple exponential space and so on. In general, we deal with the Young functions satisfying

lim -1-^ =1, a<n — 1. (4)

tn logiVlt-iKT11 ()

The borderline case is always a = n — 1 and for a > n — 1 we have an embedding into L^(0). For other results concerning these spaces and their Trudinger-type embeddings we refer the reader to [6, 8-14, 20].

The following theorem summarizes known versions of Moser's inequality for an embedding into single and multiple exponential spaces in the case of a bounded domain (see [5, 8, 17]).

Theorem 1.1.

Let K > 0, £ e N, n > 2 and a < n — 1. Suppose that Q C Rn is an open bounded set. Let 0 be a Young function satisfying (4).

(i) If u e WoL0(Q), then Jexp[e](K|u(x)|Y) dx < oo.

(II) We also have

r , ^ f < C(£,n,a, 0, Ln(Q),K), K<Ktna.

sup / expM(K\u(x)\y)dx -

ueW0L0(Q) JQ = OO, K > Klna-

l|Vu||L0(Q)<1 L

In the borderline case K = K£na, the validity of the Moser-type Inequality depends on the particular choice of the Young function 0 satisfying (4) (see [5, 17]).

If Q is not bounded, then there also exist versions of the Trudinger embedding and the Moser inequality. In such a case, one has to control ||u|l*(q> and subtract a suitable part of the Taylor expansion from the function exp^ in the integrand. More precisely, if

exP[£] t = YL cJfi

is the Taylor expansion of the function exp[£], we define

Sena (t) = aJtJ-

0<j<n/Y

In addition, let us suppose that there is C > 0 such that

^ < 0(t) < Ctn for t e

o.C l. (5)

Then we have the following result from [2].

Theorem 1.2.

Let K > 0, £ e N, n > 2, a < n — 1 and P > 0. Suppose that Q C Rn is an open set. Let 0 be a Young function satisfying (4) and (5).

(i) If u e WoL0(Q), then J [exp[e](K|u(x)|Y) - Sena(K|u(x)|Y)] dx <

(ii) We also have

up f [exp[e](K|u(x)|Y) - Sena(K|u(x)|Y)] dx

sup I |expm(K|u(x)|y) - Se,

uGWoL0(n),|Vu|L0(Q)<1 -|u|L0(Q)<P

< C(£, n, a, 0, Ln(Q),K,P) when K < K£na, = oo when K > K£na.

Again, the validity of the Moser-type inequality in the case K = K£na depends on the choice of 0.

Existence results

Let us give a brief summary of the existence results concerning problem (1). These results come from [1-4]. We distinguish two cases concerning the set 0: the case of 0 bounded and the case of 0 = Rn. We also distinguish two cases concerning the nonlinearity f(x, t): the case of the critical growth and the case of the sub-critical growth.

We start with the assumptions concerning 0. In all four cases we suppose that (4) and (5) are satisfied. In the case of the critical growth, we also assume that

0 is such that the version of Moser's sequence constructed in [5, Section 4] belongs to the unit ball in WOL0(B(R)) (with respect to the Dirichlet norm).

Let us also note that in the paper [1], there is assumed that the function t ^ 10'(t) is a Young function. However, this assumption was used in the proof of the multiplicity result only. Therefore, being interested just in the existence of nontrivial weak solutions to (1), we do not need such assumption.

Now we give a list of assumptions concerning the potential V. In all four cases we suppose that

V: 0 h-> [0, oo) is continuous. (7)

In the case of a bounded domain we also suppose that there is V1 > 0 such that

V(x) < V for all x e 0. (8)

In the case of 0 = Rn, the situation is different since we do not have the equivalence of the standard Sobolev-type norm and the Dirichlet norm for the space WL0(Rn). Therefore, in this case, problem (1) is solved with respect to the space

u G WL0(R") A 0(|u(x)|)V(x) dx

X (Rn) =

endowed with the norm

||u|U(R") = ||Vu||L0(R") + 11 u 1 L0(R",V (x)dx),

where the second Luxemburg norm is considered with respect to the measure V(x) dx, i.e.

•,V (x)dx) = lnf

A > 0: [ 0( JUM) V (x) dx < 1 Jr" \ A I

In this case we assume that the potential V satisfies (7), there is V0 > 0 such that

V(x) > V0 for all x G R" and V(x) ^ j as |x| ^00. (9)

The nonlinearity f: Q x R ^ R is supposed to satisfy the following conditions. We start with four conditions common for all four cases. Suppose

f is uniformly continuous on Q x [—10, t0] for every t0 > 0, f(x, 0) = 0 and tf(x,t) > 0 for all x G Q, t = 0, ^

there are M > 1, tM > 0 such that

0 < F(x,t)= f f(x,s) ds < M|t\]—VM|f(x,t)|, t >tM, x G Q, (11)

there is A e (0, A0) (the constant A0 > 0 is given in [1] and depends on the choice of 0) such that

F(x,t) < Af(x, t) t, t = 0, x e Q,

Urn sup (x, I. < 1 uniformly on Q, (12)

CS0(\t\)

where CS > 0 is determined in [1] (CS depends on the choice of 0).

Next, the case of the critical growth on a bounded domain means that we suppose that there are Cb > 0 and b > 0 such that

\f(x, t)\ < Cb exp[£](b\\), t e R, x e Q. (13)

In the case of the sub-critical growth on a bounded domain we suppose that for every b > 0 there is Cb > 0 such that

\f(x, t)\ < Cb exp[£](b\\), t e R, x e Q .

The critical growth on Rn means that there are Cb > 0 and b > 0 such that

\f(x,t)\< Cb\t\n—1+ Cb(expM(b\\) — Sena(b\t\Y)), t e R, x e Rn.

In the case of the sub-critical growth on Rn we suppose that for every b > 0 there is Cb > 0 such that

\f(x,t)\< C\t\n—1+ Cb(expM(b\t\Y) — Sena(b\t\Y)), t e R, x e Rn.

In this case, the first constant C is independent of b.

Finally in both cases with the critical growth we also suppose that

Urn Inf-(,[, \ , > 0 uniformly on Q.

expM(b\t\Y)

Now we can state the existence results.

Theorem 1.3.

Let e e N, n > 2, a < n — 1 and let Q C Rn be a domain. Suppose that the C1 -Young function 0: [0, oo) -> [0, to)

satisfies (4) and (5). Suppose that h e (L0(Q))* and V: Q ^ [0, to) satisfies (7). Let f: Q x R ^ R be a function

satisfying (10), (11), and (12).

critical growth for a bounded domain If, in addition, Q is bounded and we have (6), (8), (13) and (17), then there is j0 > 0 such that for every j e [0, j0), problem (1) has a nontrivial weak solution in W0L0(Q).

sub-critical growth for a bounded domain If, in addition, Q is bounded and we have (8) and (14), then there is j0 > 0 such that for every j e [0, j0), problem (1) has a nontrivial weak solution in W0L0(Q).

critical growth on Rn If, in addition, Q = Rn and we have (6), (9), (15) and (17), then there is j0 > 0 such that for every j e [0, j0), problem (1) has a nontrivial weak solution in X(Rn).

sub-critical growth on Rn If, in addition, Q = Rn and we have (9) and (16), then there is j0 > 0 such that for every j e [0, j0), problem (1) has a nontrivial weak solution in X(Rn).

The results for j e (0, j0) come from [1], while the case of j = 0 was studied in [2]. By a weak solution to (1) we mean a function u e W0L0(Q) (or u e X(Rn)) such that

jT (0'(\Vu\) \Vu\Vv + V(x)0'(\u\) u v — f(x, u)v — jh(x)y) dx = 0

for every test function v from W0L0(Q) (or X(Rn)).

Let us emphasize that the result in the sub-critical case is not just a weaker result contained in the result concerning the critical growth. Indeed, in the case of the critical growth, we admit nonlinearities with larger growth but we also have restrictive condition (17) which (together with (13) and (15), respectively) means that the growth is very special. In the case of the sub-critical growth, we have a bit more restrictive upper bound, but there is no lower bound of the absolute value of f(x, t).

New results

Now, we can formulate the main result of this paper.

Theorem 1.4.

Let £ e N, n > 2, a < n — 1. Suppose that 0: [0, oo) ^ [0, oo) is a C1 -Young function satisfying (4) and (5). Suppose that V: 0 ^ [0, to) satisfies (7). Let f: 0 x R ^ R be a function satisfying (10). Let h e tL0(0))* and v > 0.

case of a bounded domain If, in addition, 0 C Rn is a bounded domain and we have (8) and (13), then every weak solution to (1) in W0L0(0) belongs to Lto(0).

case of 0 = Rn If, in addition, 0 = Rn, 0(t) > Ctn for every t > 0 (that is £ > 2 or a > 0) and we have (9) and (15), then every weak solution to (1) in X(Rn) belongs to LTO(Rn).

Notice that in the case of a bounded domain, the assumptions of Theorem 1.4 are much more permissive than assumptions of Theorem 1.3. In the case of 0 = Rn, there is a new assumption 0(t) > Ctn (some comments concerning this assumption are given below), the assumptions concerning f(x, t) are again more permissive than assumptions of Theorem 1.3.

Let us also give some comments concerning the proof of Theorem 1.4. The basic strategy of the proof is standard. We test the weak formulation (18) with slightly modified large powers of the weak solutions. Then we apply the Moser iteration. In the case when 0(t) > Ctn (i.e. we have £ > 2 or a > 0), it is possible to estimate 0'(|Vu+|)|Vu + | (here u+ = max {u, 0}) by C|Vu+|n from below. This enables us to deal with the modular in the Sobolev space W0l,n(0) and the proof is rather standard.

In the case when £ = 1 and a < 0 we have to overcome several technical difficulties related to the fact that in a general Orlicz space, the relation between the modular and the norm is much more complicated (in comparison to the Lebesgue space-case, when the first quantity is just a power of the second one). In particular, the problems when estimating 0'(|Vu + |)|Vu+| are overcome via dealing with the truncated function u = max{u, 1} instead of u + . However, working with the function u results in the loss of the non-negativity of the second term on the left hand side of estimate (18) (that is the third term on the left hand side of (42)). Moreover, since we cannot control this term from below in the case when V(x) is not bounded from above, we have to restrict ourselves to the case of 0 bounded provided £ = 1 and a < 0 (the boundedness of V(x) is ensured by (8)). Let us also note that we cannot assume that V(x) is bounded in the case 0 = Rn, since it would lead to the loss of the Sobolev-type embedding which is necessary for the Moser iteration.

2. Boundedness of weak solutions

This section is devoted to the proof of Theorem 1.4. We distinguish two cases.

Case 0(t) > Ctn

Proof Of Theorem 1.4 if 0(t) > Ctn. We start with some computations common for both cases concerning 0 (0 is either bounded or 0 = Rn). Assume that u is a weak solution to (1).

Fix L > 0 and ¡3 > 1. Let us note that in the main part of the proof, 3 is replaced by ¡¡j, j e N, from a suitable sequence of parameters {¡j}. These ¡¡j are bounded away from 1, thus we can use the estimate 3 — 1 > C (by C we denote a generic positive constant independent of j, this constant may vary from expression to expression as usual). We further define u+ = max {u, 0} and u+ = min {u+, L}. Let us set

y = u+tu+)n(3—1). (19)

Since u+ is bounded and ¡3 > 1, we can see that y can be considered as a test function in (18) (see [15, Theorem 7.8]). We also obtain that almost everywhere

Vy = Vu+(u+)n(3—1) + n(3 — 1)Vu+tu+)n(3—1). (20)

Indeed, on the set {Vu+ = 0} we have u+ = u+ almost everywhere. Similar observations are several times used also in the rest of the paper (without warning).

Now, testing (18) with the function v given by (19), we obtain with the aid of (20)

|j0'(\Vu + \)\Vu+\(u+)n(^—1)+ n(fi — 1)0'(\Vu+\)\Vu+\(u+)n^1) + V (x)0'(u+)u+(u+)n^1))

= J (f(x,u)u+(u+)n(^1) + jh(x)u+(u+)nW—1)) dx < J^\f(x,u)\u+(u+ )n(fi—1) + j\h(x)\u+(u+)n(fi—1)) dx.

Hence, as 0'(t) > Ctn—1 (we have 0(t) > Ctn for all t > 0 and the fact that 0 is a Young function implies 10'(t) > 0(t) for all t > 0), we obtain

|J(\Vu + \(u+f—1 )n + n(fi — 1)(|Vu+|(u+)^1 )n + V(x)(u+ (u+f—1 )n] dx

< Cj [\f(x,u)\u+(u+)nW—''»+ j\h(x)\u+(u+ )nW—''»] dx.

Next, we define an auxiliary function

Wl = u+( u+ f—\

We see that we have almost everywhere

VWl = Vu+(u+)P—1 + (fi — 1)Vu+(u+)i—1. Furthermore, by (23), we have the estimate

^ \ VWl \ ndx < 2n|j( \ Vu+\(u+)^—1)n + (fi — 1)n (| Vu+1 (u+)^—1)"] dx

< Cfin—1 fj( \ Vu+\(u+)fi—1)n + n(fi — 1)(|Vu+|(u+)fi—1)n] dx.

Hence, from (21), (22) and (24) we obtain

J\ VWl \n + V (x)WnLdx < Cfin—1f^ [ \ f (x, u) \ u+(u+)n(fi—1) + j \ h(x) \ u+(u+)nfi—1)] dx.

In the rest of the proof we distinguish two cases.

Case of a bounded domain. First let us fix s > 1. Using the facts that V(x) is nonnegative, f(x, u) e L(2n) (Q) (by (13) and Theorem 1.1 (i)), h(x) e (L^Q))* C L(2n) (Q) (indeed, the dual space to L0(Q) corresponds to the associated Young function to 0, and since 0 "behaves similarly" as the function t ^ tn, the associated Young function "behaves similarly" as the function t ^ tn ), Holder's inequality and (25) we arrive at

j \VWL\n dx < Cfin—1 j [ \f(x, u)\u+(u+)n(fi—1) + j\h(x)\u+(u+)n(fi—1)] dx < Cfin—1 |u+(u

+ /,,+ )n«?—1>!

L2n(Q)

Hence, by the continuous embedding of W0,n(Q) into L2sn (Q) ( see ( )), 0 < u+ < u+ and ( ), we have

IL2sn2(Q)

< C||VWL||nn,0, < Cfin—1|u+(u+)

n(fi—1)|L2n(n) < Cfin— |(u+)1+n'fi—1' |L2n(n).

L2sn2(Q)

This implies by the Lebesgue Monotone Convergence Theorem

||Kf1^2,0, < C3n—1'||(u+r<3—1'>||L2n(n) = C3n—1tWuXinH-l1))(0). Therefore, as (1 + n(fi — 1)/(fin) < 1, we have

IKIUn2(n) < C^»^Wu+ WL+^Zn < CV3PV3 max{1, Wu+WL2n(1+n(3—1))(0)}. (26)

Now, we are going to iterate inequality (26). We define the sequence {¡j} by

¡1 = s and 2sPjn2 = 2n t1 + n(3J+1 — 1)), j e N, the second property can be also written as ¡¡+1 = sfij + (n — 1)/n. Next, we observe that

¡j > sj — to, j oo, (27)

p| c 1/fj < C(28) j=1

(we can suppose that C > 1) and

=exp Г ПГ < exp C + £ ^ =exp C + log s . £ I ) < то (29)

j=i j=i ßj \ j=i I \ j=i

(indeed, the function t — (log t)/t s decreasing on (exp 1, to) and we have the estimate ¡j > sj, j e N). Therefore we obtain from (26), (28) and (29), j

\U + 11^2 , < П Cl/ßjTlß j KWp} < C, j G N.

j=1 j = 1

Thus (27) implies that u+ e LTO(0). In the same way we obtain that u— = min {u, 0} e LTO(0) and hence u e LTO(0). Case of 0 = Rn. In this case, we can see that the assumption h e (L0(Rn))* and assumptions (4) and (5) imply

hX{|h|<1} e Ln(Rn) and hX{|h|>1} e L<2n>'(Rn) (30)

(here X{|h|<i} is the characteristic function of the set {|h| < 1}, similarly for X{|h|>i}). Next, for given £ e N, n e N, a < n — 1 and p > 1, one can easily prove that there is C > 1 such that

texp[£](t) — Sena(t))p < CtexpM(pt) — Sena(pt)), t > 0. (31)

Therefore we have f (x, u) e L(2n) (0) (indeed, it is enough to use (15), Theorem 1.2 (i) and (31)). Now, we can use this fact, (30) and Holder's inequality to obtain

f (^иЖКГ-1^ИМ*)1 и+ЫУ('-1)) dx

M" v '

< C ||K)W)||l2»(r») + (C ||Kr(ß-1>\\LnR") + C ||КГ^-1)Ь„ R)).

Hence, Inequality (25) turns to

(1^11^) + \\WL\\LnRn,vwx))n < cr-1(||(u+)1+^-

,+ \1+n(P-

Next, the norm on the left hand side corresponds to a space continuously embedded into Lq(Rn) for every q G [n, to) (cf. [2, Proposition 2.10]), and thus we have for s > 2 fixed

_ ■2(Rn) < C(\VWL\Ln(Rn) + || WLWLnpnv,),,,))n

< C^n-1(|(a + )1+n(^-1)|Ln(Rn) + ||(" + )W||L2n(Rn))

which implies by the Lebesgue Monotone Convergence Theorem,

Cpn-1(U(u+y+n(- ||Ln(Rn) + H^)1"^^ )).

LsPn2 (Rn )

< CPnl \\ „+ \1+n(^-1) + \\„+«nn(l3-t)

l\\U WLn(1+n(P-1))(Rn) + \\U WL2n(1+n(P-1))(Rn)

Now, let us define the sequence {fij} by

01= s and sfijn2 = n (1 + n(fij+1 — 1)), j e N, i.e. j8j+i = sfij + (n — 1)/n. We observe that fi > sj ^ to, j ^ to. Next, we apply the interpolation inequality

Hv y (Rn) <Hv fLP Rn )|v ||L—(Rn), where p<r<q, L = ^ l—P , 1 — L = 1 ,

to obtain from (32) and sfij—1n2 < 2sfi—1n2 < sfijn2 (recall that fi > sfij—1 > 2fi—1)

r q - p

„Pin

< Cpn \U

< Cpn \U

\\1+n(pj-1) + \\U+\\1+n(pj-1) \

^-^(Rn) + \\U WL^j-m2^))

\\Un(Pj-1) + \\„+W eJ (1+n(Pj-1))W„+W(1- eJ )(1+n(Pj-1)) )

^-^Vn) + WU W2™) \\U ^^(Rn) )

1 spn2 - 2spMn2 1 Pj - 26

1 s - 2

j 2 sfijn2 — sfij—m2 2 fi — fij—1 Notice that for fixed j e N, (33) is an inequality of the type

2 s - 1

j —> oo.

Af < Cpn (A-P + tflfA^-9)anp), a,9 G (0,1).

This inequality in the case Aj—1 > Aj implies Af < 2CfinAa—, while in the case Aj—1 < Aj we have

Anfi—(1—6)anfi < 2CfinALanfi

Therefore the following estimate is valid in both cases (we can suppose that C > 1)

Aj < (2Cfi)1/(fi(1—(1—L>a»i

1 max jA^A-1-™}.

Hence, as L is bounded away from zero (see (34)) and as a < 1, we can use an obvious inequality La < 1 — (1 — L)a, a,L e [0,1], to obtain Aj < (2Cfi)C/fi max {1,Aj—1}. Therefore (32) takes the form

< (2CPJ)C/PJ max{1, \\u+

<<Lspj-1n

)}, j . 2.

This is an estimate of the same type as (26) and thus we can conclude the proof in the same way as in the case of a bounded domain. □

Case of £ = 1, a < 0 and Q being bounded.

In this case we have to start with some preliminary work. We need a suitable estimate of the modular by a norm In the Orlicz space L0(Q).

Lemma 2.1.

Let £ = 1, n > 2, a < 0 and suppose that the Young function 0 satisfies (4) and (5). Then there is s0 > 1 such that if s > s0, then

[0( |v |) dx < sn ||l*(Q) < s log I a's.

Consequently, if s > 0, then

\0(|vI) dx < sn Hv|Uq> < max{s,s0} log 1 a|(max{s,s0}).

Proof. Suppose that we have

\ 0( I v I) dx < sn (35)

for s very large (the lower bound s0 is specified below). In view of the definition of the Luxemburg norm on L0(Q), our aim is to show that

'J^vv\\dx < 1.

Q \s log1 a s

First, let us define an auxiliary function 0(t) = t" loga(e + t), t > 0. From (4) and (5) we observe that there is C > 1 such that

1 0(t) < 0(t) < C0(f), t > 0. (36)

Therefore we have for s large enough

[ 0(^L)dx < C [ 0 ^ —Vj—) dx

Jq \s log 1 a s/ - Jq \s log 1 a (37)

CilvI" loga(e + 1 V| , ) dx < -(I1+ ¡2),

"1 ajs Jq 1 1 M s log 1 alsl ~ sn log" 1 a js 2)

s" log" 1 a 1 s Jq \ s log 1 alsl ~s" log" 1 a js where

¡1=1 ,, Ivl" loga(e + -jV\-)dx, ¡2= ( | vj" loga(e + dx.

j{jv I>(s log|a|s)2} \ s log| | si J{IvI<s3} \ s logM si

Now, let us estimate I1 and I2. Using a < 0, inequality log(e + y/i) > (1/2)log(e +1), t > 0, (35) and (36) we obtain

¡1=1 |v|" loga(e + ^dx <[ |v|" loga(e + VM) dx

J{IvI>(s log|a|s)2} \ s log| | ^ / J{Iv|>(slog|a|s)2}

< 2|a| \ I vI" loga(e + |v|) dx = C f 0(|v|) dx < C \ 0(|v|) dx < Cs". Jq Jq Jq

Next, from < 0 and (35) we have for s large enough,

|2 =f IvI" log a (e + |v| ) dx <[ IvI" dx = log|a|(e + s3)f IvI" loga(e + s3) dx

J{IvI<s3} \ s log| a |W J{IvI<s3} J{IvI<s3}

I <s3} \ s logI a Isl J{IvI<s3} J{IvI<s3}

< log|a| (e + s3^ |vI" log a(e + |v|) dx (39)

J{M<s3}

= log|a|(e + s3) \ 0(|v|) dx < C log|a|s \ 0(|v|) dx < Cs" log|a|s.

J{IvI<s3} Jq

Finally, from n > 2, (37), (38) and (39) we obtain for s large enough,

\v\ \ , C ,, ,, Csn Log\a\s „ ,_,

, 0 —vr" dx < -¡-p- (I1 + I2) < -< 1. □

Jn \s Log \ a\sj ~ sn Logn\as 2 sn Logn \a

Now, we can prove Theorem 1.4 also in the remaining case.

Proof of Theorem 1.4 in the case e = 1, a < 0 and Q bounded. Assume that u is a weak solution to (1). Fix L > 0 and fi > 1. We define u = max{u, 1} and uL = mln {u, L}. Let us set

--n(fi—1)

— II II ^ '

Since uL is bounded and fi > 1, we can see that v can be considered as a test function in (18). We also observe that we have almost everywhere

Vv = VUULnfi—1) + nfi — 1) VuLunL{fi—1). (41)

Now, testing (18) with the function v given by (40), we obtain with the aid of (41),

0'( \ u \) -

J^0'(\Vu\)|vu|uLn(fi—1) + n(fi—1)0'(|vul|)|vul|uLn(fi—1) + v(x)0'( \u\)UuuLnfi—1)) dx

jjf(x,u)UULn(fi—1) + jh(x)UrLifi—1)) dx.

Using the facts that V(x) is bounded (by (8)), 0'( \u\) e L<2n>'(Q) (since W0L0(Q) is embedded into Lq(Q) for every q e [1, to)), f(x, u) e L(2n)'(Q) (by (13) and Theorem 1.1 (i)), h(x) e (L0(Q))* C L<2n>'(Q) (indeed, the dual space to L0(Q) corresponds to the associated Young function to 0, and since 0 "behaves similarly" as the function t h-> tn, the associated Young function "behaves similarly" as the function t h-> tn ) and Holder's inequality we arrive at

J 10'( \ VU \) \ VU\dnLifi—1) + nfi — 1) 0'( \ vul \) \ vul \ dx

< J (\f(x, u)\UULn(fi—1) + j\h(x)\UULn(fi—1) + V(x)0'( \u\)UULn(fi—1)) dx (42)

< C |uUL(fi ' |L2n(Q) < C ||u|L+>(1<+"(fi—1))(Q).

Next, we define the function WL = uuf—1. We observe that almost everywhere

vwl = VUUfi—1 + (fi — 1) VULUfi—1.

Similarly as in the proof of Lemma 2.1, we define an auxiliary function 0(t) = tn Loga(e +1), t > 0. From (4) and (5) we observe that there is C > 1 such that

10(t) < 0(t) < C0(t), t > 0. (43)

Next, as the function 0(t) satisfies the A2-condition and the inequality 0(st) < Csn0(t), t > 0, s > 1 (both properties are easily verified by a short computation), we obtain

0(\vwl\) < C0( \vwl\) = C0(|VUUfi—1 +(fi — 1)VULUf-11)

< C0(|VUUfi—1|) + C0(|(fi — 1)VU^^'D < C0(|VUUfi—1|) + C(fi — 1)n0(|VULUf—1|) = C\VU\nUnLifi—1) Loga(e + \ VU\Ufi—1) + Cfi — 1)n\ VUL\nUnLifi—1) Loga(e + \ VUL\Ufi—1).

Furthermore, uL > 1, a < 0, (43) and the estimate 0(f) < t0'(t), t > 0, yield

0(|V WL|) < C\vu\nunifi-1) log"(e + |VU|) + C(fi- ^V^ ^uf- log"(e + |VuzJ)

= C 0 (|Vu|) un(^-1) + C (fi - 1)n 0 (|vui|)un(^-1) (44)

< C0(|Vu|)un(fi-1) + C(fi- 1)n0(|VUi|)Un(fi-1)

< C0'(|VU|)|VU|Un(fi-1) + C (fi - 1)n0'(|VUi|)|VUi|un(fi-1).

Therefore from (42) and (44) we obtain

f 0(|VWL|) dx < C(fi - 1)n-1\\u\\l Jn

+n(fi-1)

L2n(1+n(fi-1)) (o).

Now, let us fix s > 2. To simplify our notation, let us write q = \\u\\L2n(1+nfi-1))(0). Using the continuous embedding of W0L0(O) into L2sn2(0) and Lemma 2.1 we obtain from the above

IUUfi-1||L2sn2(n) < C ma^ so, (Cfin-1Q1+nfi-1>)1/^ log|a|( ma^ so, (Cfin-1 Q1+n<fi-1> ).

This implies by the Lebesgue Monotone Convergence Theorem, the fact that fi is bounded away from 1, (1+ n(fi - 1))/n < fi, and by some trivial estimates

\\u\\fi . = llufiH^ < mav;c Cft(n-1)/nQ(1+n(fi-1»/n\ftiog|a|

\ \\L2sfin2 (0) _ I r llL2sn2(0)

2,™ < ma4C, Cfin-1>/Y+nfi-1»/n}Je log|a|(max{Cfiq}) < max{C,CP2Qfi}.

u\L2sfin2(o) < Cmax{1,fi2/fi\u\L2n(1+n(fi-1)),n)}.

This is a version of estimate (26) for which we can follow the steps from the proof in the case 0(f) > Ctn to obtain that u G L^(O). In the same way we can show that min {u, -1} e L^(O) and thus we have u e L^(O). □

Acknowledgements

The author would like to thank Professors Andrea Cianchi, Pavel Drabek and Stanislav Hencl for fruitful discussions. The author was supported by the ERC CZ grant LL1203 of the Czech Ministry of Education.

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