A priori bounds for weak solutions to elliptic equations with nonstandard growth

In this paper we study elliptic equations with a nonlinear conormal derivative boundary condition involving nonstandard growth terms. By means of the localization method and De Giorgi's iteration technique we derive global a priori bounds for weak solutions of such problems.


Introduction
The present paper is concerned with global a priori bounds for elliptic equations with nonlinear conormal derivative boundary conditions which may contain nonlinearities with variable growth exponents. More precisely, let Ω be a bounded domain in R N , N > 1, with Lipschitz boundary Γ := ∂Ω and let p ∈ C(Ω) be a function that satisfies 1 < p − := inf Ω p(x). We deal with elliptic equations of the form − div A(x, u, ∇u) = B(x, u, ∇u) in Ω, A(x, u, ∇u) · ν = C(x, u) on Γ, (1.1) where ν(x) denotes the outer unit normal of Ω at x ∈ Γ, and A, B and C satisfy suitable p(x)-structure conditions, see (H) below. An important special case of (1.1) which fits in our setting is given by −∆ p(x) = B(x, u, ∇u) in Ω, |∇u| p(x)−2 ∂ ν u = C(x, u) on Γ.
Here the operator div A becomes the so-called p(x)-Laplacian ∆ p(x) u = div(|∇u| p(x)−2 ∇u), which reduces to the standard p-Laplacian if p(x) ≡ p.
In recent years there has been a growing interest in the study of elliptic problems with a p(x)-structure, which are also termed problems with nonstandard growth conditions. Equations of this type appear in the study of non-Newtonian fluids with thermo-convective effects (see Antontsev and Rodrigues [4], Zhikov [35]), electrorheological fluids (see Diening [8], Rajagopal and Růžička [29], Růžička [31]), the thermistor problem (see Zhikov [36]), or the problem of image recovery (see Chen et al. [5]).
Throughout the paper we impose the following conditions.
for a.a. x ∈ Ω, and for all s ∈ R, and all ξ ∈ R N . Here a i , b j and c l are positive constants, p ∈ C(Ω) with inf Ω p(x) > 1, and q 0 ∈ C(Ω) as well as q 1 ∈ C(Γ) are chosen such that with the critical exponents holds for all nonnegative test functions ϕ ∈ W 1,p(·) (Ω), where dσ denotes the usual (N − 1)-dimensional surface measure. This definition makes sense, since thanks to assumption (H) the integrals in (1.2) are finite, by Hölder's inequality and embedding results for W 1,p(·) (Ω)-functions, see below.
The main goal of this paper is to prove a priori bounds for weak sub-and supersolutions, in particular for weak solutions of problem (1.1). Using the notation y + = max(y, 0), our main result reads as follows.
(i) If u ∈ W 1,p(·) (Ω) is a weak subsolution of (1.1) then Note that the constants a 0 , a 1 , a 2 , which appear in (H1), do not play any role in determining the constants α and C. The finiteness of the right-hand sides in (i) and (ii) is a consequence of the compact embedding W 1,p(·) (Ω) → L q0(·) (Ω) and the fact that the trace operator is a bounded operator from W 1,p(·) (Ω) into L q1(·) (Γ) (see Fan et al. [15,Theorem 1.3] and Fan [12,Corollary 2.4]). We further point out that we merely assume continuity for the variable exponents p, q 0 , and q 1 ; log-Hölder continuity conditions are not required.
Our proof of Theorem 1.1 uses De Giorgi's iteration technique and the localization method. By means of the latter we are able to reduce the estimates involving variable exponents to ones with constant exponents, which then also allows us to apply classical embedding results. This crucial step in the proof is achieved by means of an appropriate partition of unity.
By the definition of sub-and supersolution of (1.1) one easily verifies that a weak solution is both a weak subsolution and a weak supersolution. Hence we have the following.
Corollary 1.2. Let the assumptions (H) be satisfied and let u ∈ W 1,p(·) (Ω) be a weak solution of (1.1). Then u ∈ L ∞ (Ω) and the estimates in (i) and (ii) from Theorem 1.1 are valid.
The main novelty of the paper consists in the generality of the assumptions needed to establish the boundedness of weak solutions to (1.1). In particular the assumptions on the nonlinearity C are rather general, allowing for a growth term with variable exponent, which seems to be optimal. Another novelty is the use of the localization technique in the context of global a priori estimates for problems with variable exponents and nonlinear conormal derivative boundary conditions.
Let us comment on some relevant known results on elliptic problems with p(x)structure. Local boundedness of solutions to the equation has been studied by Fan and Zhao [16]. There it is shown that under suitable structure conditions every weak solution u of (1.3) (corresponding to test functions ϕ ∈ W 1,p(x) 0 (Ω)) belongs to L ∞ loc (Ω), and if in addition u is bounded on the boundary Γ, then u ∈ L ∞ (Ω). The proof uses De Giorgi iterations as well. Recently, Gasiński and Papageorgiou (see [19,Proposition 3.1]) studied global a priori bounds for weak solutions to the equation in Ω, where the Carathéodory function g : Ω × R → R satisfies a subcritical growth condition. They proved that every weak solution u ∈ W 1,p(·) (Ω) of problem (1.4) belongs to L ∞ (Ω) provided p ∈ C 1 (Ω) satisfying 1 < min x∈Ω p(x). L ∞ -estimates for solutions of (1.1) in case p(x) ≡ p with q 0 (x) = q 1 (x) ≡ p have been established by the first author in [33,34] following Moser's iteration technique (for constant p see also Pucci and Servadei [28]).
The paper is organized as follows. In Section 2 we fix some notation and recall the definition of the variable exponent spaces L p(·) (Ω) and W 1,p(·) (Ω). We further state a lemma on sequences of numbers which will be needed for the De Giorgi iterations. The main result is proved in Section 3. From the structure conditions we first derive truncated energy estimates. These are then used, together with the localization method and embedding results, to prove suitable iterative inequalities, which in turn imply the desired a priori bounds.

Notations and preliminaries
Suppose that Ω is a bounded domain in R N with Lipschitz boundary Γ and let p ∈ C(Ω) with p(x) > 1 for all x ∈ Ω. We set p − := min x∈Ω p(x) and p + := max x∈Ω p(x), then p − > 1 and p + < ∞. By L p(·) (Ω) we identify the variable exponent Lebesgue space which is defined by The variable exponent Sobolev space W 1,p(·) (Ω) is defined by For more information and basic properties of variable exponent spaces we refer the reader to the papers of Fan and Zhao [17], Kováčik and Rákosník [22] and the recent monograph of Diening et al. [10]. If p(x) ≡ p is a constant, the usual Sobolev space W 1,p (Ω) is endowed with the norm For q 0 ∈ C(Ω) and q 1 ∈ C(Γ) (as in (H)) we define For s ∈ [1, ∞) we further use the notation The following lemma concerning the geometric convergence of sequences of numbers will be needed for the De Giorgi iteration arguments below. It can be found, for example in [32]. The case δ 1 = δ 2 is contained in [ 3. Truncated energy estimates and proof of Theorem 1.1 Our proof of the sup-bounds for weak subsolutions of (1.1) is based on the following lemma on truncated energy estimates.
as u q0(x) > u > 1 in A k . Now, we are going to estimate the right-hand side of (3.1). By Young's inequality with ε ∈ (0, 1] and condition (H3) we have Thanks to condition (H4), the boundary integral can be estimated through as u > 1 on Γ k . Combining (3.1)-(3.4) and choosing ε = min(1, a3 2b0 ) gives Dividing the last inequality by a3 2 > 0 yields the assertion of the lemma.
The corresponding result for supersolutions reads as follows.
Lemma 3.2. Let the conditions in (H) be satisfied. If u is a weak supersolution of (1.1), there holds and d 1 and d 2 are the same constants as in Lemma 3.1.
Proof. The proof is analogous to the previous one. We take ϕ = −(u + k) − = − min(u + k, 0) ≥ 0 as test function in (1.2), which now holds with the '≥ '-sign, and use the same arguments as in the proof of Lemma 3.1. This yields the asserted inequality.
Now we are in position to prove the main result of this paper.
Proof of Theorem 1.1.
(i) Definition of the iteration variables Z n ,Z n , and basic estimates. Let now k n = k 2 − 1 2 n , n = 0, 1, 2, . . . , with k ≥ 1 specified later and put We have
(iv) Estimating Z n+1 . Next we want to derive a suitable estimate for the term Z n+1 from above. To this end, we make again use of the partition of unity introduced in step (ii). We have Let now i ∈ {1, . . . , m} be fixed, and suppose that r ∈ {q − 0,i , q + 0,i }. Then p − i ≤ r < (p − i ) * and r ≤ q + , where q + = max(q + 0 , q + 1 ). By Hölder's inequality and the continuous embedding , we may estimate as follows.
(vi) The iterative inequality for Y n . Recall that Y n = Z n +Z n . Hence (3.17) and (3.22) yield n + Y 1+δ2 n with K = max(d 7 , d 11 ), b = max(d 8 , d 12 ),η = max(η,η), and where 0 < δ 1 ≤ δ 2 are given by Without loss of generality we may assume that b > 1. Now we may apply Lemma 2.1, which says that Y n → 0 as n → ∞ provided Tracing back the constants, we see that the first part of the theorem is proved. The supersolution case can be done analogously, replacing u with −u and A k withÃ k , and using Lemma 3.2 instead of Lemma 3.1. This completes the proof.