Nonlinear elliptic boundary value problems at resonance with nonlinear Wentzell boundary conditions

In the first part of the article, we give necessary and sufficient conditions for the solvability of a class of nonlinear elliptic boundary value problems with nonlinear boundary conditions involving the q-Laplace-Beltrami operator. In the second part, we give some additional results on existence, uniqueness and continuous dependence estimates of the weak solutions for these classes of nonlinear problems. Finally, we show that these solutions are also bounded and provide explicit estimates.

In a recent paper [12], the authors have formulated necessary and sufficient conditions for the solvability of (1.1) when p = q = 2, by establishing a sort of "nonlinear Fredholm alternative" for such elliptic boundary value problems. We shall now state their main result.
Defining two real parameters λ 1 , λ 2 ∈ Ê + by , (1.2) this result reads that a necessary condition for the existence of a weak solution of (1.1) is that while a sufficient condition is where R(α j ) denotes the range of α j , j = 1, 2 and int(G) denotes the interior of the set G. Relation (1.3) turns out to be both necessary and sufficient if either of the sets R (α 1 ) or R (α 2 ) is an open interval. This particular result was established in [12,Theorem 3], by employing methods from convex analysis involving subdifferentials of convex, lower semicontinuous functionals on suitable Hilbert spaces. As an application of our results, we can consider the following boundary value problem −∆u + α 1 (u) = f (x) , in Ω, b (x) ∂ n u = g (x) , on ∂ Ω, (1.5) which is only a special case of (1.1) (i.e., ρ = 0, α 2 ≡ 0 and p = 2). According to [12,Theorem 3] (see also (1.4)), this problem has a weak solution if which yields the result of Landesman and Lazer [17] for g ≡ 0. This last condition is both necessary and sufficient when the interval R (α 1 ) is open. This was put into an abstract context and significantly extended by Brezis and Haraux [8]. Their work was much further extended by Brezis and Nirenberg [9]. The goal of the present article is comparable to that of [12] since we want to establish similar conditions to (1.4) and (1.6) for the existence of solutions to (1.1) when p, q = 2, with main emphasis on the generality of the boundary conditions.
Our first main result is as follows (see Section 4 also). for some constants C j > 1, j = 1, 2. If u is a weak solution of (1.1) (in the sense of Definition 4.10 below), then (1.8) then (1.1) has a weak solution.
Our second main result of the paper deals with a modified version of (1.1) which is obtained by replacing the functions α 1 (s) , α 2 (s) in (1.1) by α 1 (s) + |s| p−2 s and α 2 (s) + ρb |u| q−2 u, respectively, and also allowing α 1 , α 2 to depend on x ∈ Ω. Under additional assumptions on α 1 , α 2 and under higher integrability properties for the data ( f , g), the next theorem provides us with conditions for unique solvability results for solutions to such boundary value problems. Then, we obtain some regularity results for these solutions. In addition to these results, the continuous dependence of the solution to (1.1) with respect to the data ( f , g) can be also established. In particular, we prove the following Theorem 1.2. Let all the assumptions of Theorem 1.1 be satisfied for the functions α 1 , α 2 . Moreover, for each j = 1, 2, assume that α j (t) /t → 0, as t → 0 and α j (t) /t → ∞, as there exists a unique weak solution to problem (1.1) (in the sense of Definition 5.3 below) which is bounded.
We organize the paper as follows. In Section 2, we introduce some notations and recall some well-known results about Sobolev spaces, maximal monotone operators and Orlicz type spaces which will be needed throughout the article. In Section 3, we show that the subdifferential of a suitable functional associated with problem (1.1) satisfies a sort of "quasilinear" version of the Fredholm alternative (cf. Theorem 3.5), which is needed in order to obtain the result in Theorem 1.1. Finally, in Sections 4 and 5, we provide detailed proofs of Theorem 1.1 and Theorem 1.2. We also illustrate the application of these results with some examples.

PRELIMINARIES AND NOTATIONS
In this section we put together some well-known results on nonlinear forms, maximal monotone operators and Sobolev spaces. For more details on maximal monotone operators, we refer to the monographs [4,7,20,21,27]. We will also introduce some notations.
(ii) maximal monotone, if it is monotone and the operator I + A is invertible.
Next, let V be a real reflexive Banach space which is densely and continuously embedded into the real Hilbert space H, and let V ′ be its dual space such that V ֒→ H ֒→ V ′ .

Functional setup.
Let Ω ⊂ Ê N be a bounded domain with a Lipschitz boundary ∂ Ω.
For 1 < p < ∞, we let W 1,p (Ω) be the first order Sobolev space, that is, Since Ω has a Lipschitz boundary, it is well-known that there exists a constant C > 0 such that u Ω,p s ≤ C u W 1,p (Ω) , for all u ∈ W 1,p (Ω), Moreover the trace operator Tr(u) := u | ∂ Ω initially defined for u ∈ C 1 (Ω) has an extension to a bounded linear operator from Throughout the remainder of this article, for 1 < p < N, we let If p > N, one has that . For more details, we refer to [23,Theorem 4.7] (see also [19,Chapter 4]).
For 1 < q < ∞, we define the Sobolev space W 1,q (∂ Ω) to be the completion of the space C 1 (∂ Ω) with respect to the norm where we recall that ∇ Γ u denotes the tangential gradient of the function u at the boundary ∂ Ω. It is also well-known that W 1,q (∂ Ω) is continuously embedded into L q t (∂ Ω) where , for all u ∈ W 1,q (∂ Ω). For p, q ∈ [1, ∞], we define the Banach space If p = q, we will simply denote |F | p,p = |F | p . Identifying each function u ∈ W 1,p (Ω) with U = (u, u| ∂ Ω ), we have that W 1,p (Ω) is a subspace of X p,p (Ω, µ).

Musielak-Orlicz type spaces.
For the convenience of the reader, we introduce the Orlicz and Musielak-Orlicz type spaces and prove some properties of these spaces which will be frequently used in the sequel (see Section 5).

Definition 2.3.
Let (X, Σ, ν) be a complete measure space. We call a function B : The complementary Musielak-Orlicz function B is defined by It follows directly from the definition that for t, s ≥ 0 (and hence for all t, s ∈ Ê) Definition 2.4. We say that a Musielak-Orlicz function B satisfies the (△ 0 α )-condition (α > 1) if there exists a set X 0 of ν-measure zero and a constant C α > 1 such that for all t ∈ Ê and every x ∈ X \ X 0 .
We say that B satisfies the (∇ 0 2 )-condition if there is a set X 0 of ν-measure zero and a constant c > 1 such that for all t ∈ Ê and all x ∈ X \ X 0 .
• Φ is even, strictly increasing and convex; We say that an N -function Φ satisfies the (△ 2 )-condition if there exists a constant C 2 > 1 such that

Remark 2.6.
For an N -function Φ, we let ϕ be its left-sided derivative. Then ϕ is left continuous on (0, ∞) and nondecreasing. Let ψ be given by Then As before for all s,t ∈ Ê, The function Ψ is called the complementary N -function of Φ. It is also known that an N -function Φ satisfies the (△ 2 )-condition if and only if for some constant c ∈ (0, 1] and for all t ∈ Ê, where ϕ is the left-sided derivative of Φ. Proof. We have Since ϕ(ψ(s)) ≥ s for all s ≥ 0 and s/Ψ(s) and s/(s − 1) are decreasing, we get for t := ψ(s), that Hence, Ψ(t)2c ≤ Ψ(ct).

Corollary 2.8. Let B be a Musielak-Orlicz function such that B(x, ·) is an
On this space we consider the Luxemburg norm · X,B defined by

Proposition 2.10. Let B be a Musielak-Orlicz function which satisfies the
Proof. If B satisfies the (∇ 0 2 )-condition, then there exists a set X 0 ⊂ X of measure zero such that for every ε > 0 there exists α = α(ε) > 0, for all t ∈ Ê and all x ∈ X\X 0 . Let λ ∈ (0, ∞) be fixed. For ε := 1/λ there exists α > 0 satisfying the above inequality. We will show that ρ B (u) ≥ λ u X,B whenever u X,B > 1/α. Assume that u X,B > 1/α and let δ > 0 be such that α = (1 + δ )/ u X,B . Then for all n ∈ AE. If we assume that the last inequality does not hold, then and this clearly contradicts the definition of u X,B . Therefore, we must have The proof is finished.

Some tools.
For the reader's convenience, we report here below some useful inequalities which will be needed in the course of investigation.
Lemma 2.12. Let a, b ∈ Ê N and p ∈ (1, ∞). Then, there exists a constant C p > 0 such that Proof. The proof of (2.12) is included in [10,Lemma I.4.4]. In order to show (2.11), one only needs to show that the left hand side is non-negative, which follows easily.
The following result which is of analytic nature and whose proof can be found in [22,Lemma 3.11] will be useful in deriving some a priori estimates of weak solutions of elliptic equations.

THE FREDHOLM ALTERNATIVE
In what follows, we assume that Ω ⊂ Ê N is a bounded domain with Lipschitz boundary Next, let ρ ∈ {0, 1} and p, q ∈ (1, +∞) be fixed. We define the functional J ρ : 2 → [0, +∞] by setting Throughout the remainder of this section, we let µ : The following result can be obtained easily. The following result contains a computation of the subdifferential ∂ J ρ for the functional J ρ .
Let W = (w, w| ∂ Ω ) ∈ D(J ρ ), 0 < t ≤ 1 and set V := tW + U above. Dividing by t and taking the limit as t ↓ 0, we obtain that where we recall that Choosing w = ±ψ with ψ ∈ D(Ω) (the space of test functions) and integrating by parts in Therefore, the single valued operator ∂ J ρ is given by Since the functional J ρ is proper, convex and lower semicontinuous, it follows that its subdifferential ∂ J ρ is a maximal monotone operator.
In the following two lemmas, we establish a relation between the null space of the operator A ρ := ∂ J ρ and its range.
Conversely, let U = (u, u| ∂ Ω ) ∈ N A ρ . Then, it follows from (3.5) that Since Ω is bounded and connected, this implies that u is equal to a constant. Therefore, U = C1 and this completes the proof.

Lemma 3.4. The range of the operator A ρ is given by
: = Ω FV dµ.
Let us now prove the converse. To this end, let F ∈ 2 be such that It is clear that V ρ,0 is a closed linear subspace of V ρ ∩ 2 ֒→ 2 , and therefore is a reflexive Banach space. Using [18, Section 1.1], we have that the norm It is easy to see that F ρ is convex and lower-semicontinuous on 2 (see Proposition 3.1).
We show now that F ρ is coercive. By exploiting a classical Hölder inequality and using (3.8), we have Obviously, this estimate yields Therefore, from (3.9), we immediately get This inequality implies that and this shows that the functional F ρ is coercive. Since F ρ is also convex, lowersemicontinuous, it follows from [3, Theorem 3.3.4] that, there exists a function U * ∈ V ρ,0 which minimizes F ρ . More precisely, for all V ∈ V ρ,0 , F ρ (U * ) ≤ F ρ (V ); this implies that for every 0 < t ≤ 1 and every V ∈ V ρ,0 , Hence, Using the Lebesgue Dominated Convergence, an easy computation shows that and using the fact that Therefore, A ρ (U) = F. Hence, F ∈ R(A ρ ) and this completes the proof of the lemma.
The following result is a direct consequence of Lemmas 3.3, 3.4. This is the main result of this section.
Theorem 3.5. The operator A ρ = ∂ J ρ satisfies the following type of "quasi-linear" Fredholm alternative:

NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF SOLUTIONS
In this section, we prove the first main result (cf. Theorem 1.1) for problem (1.1). Before we do so, we will need the following results from maximal monotone operators theory and convex analysis. Definition 4.1. Let H be a real Hilbert space. Two subsets K 1 and K 2 of H are said to be almost equal, written, K 1 ≃ K 2 , if K 1 and K 2 have the same closure and the same interior, that is,  In particular, if the operator A + B is maximal monotone, then and this is the case if ∂ (ϕ 1 + ϕ 2 ) = ∂ ϕ 1 + ∂ ϕ 2 .

Assumptions and intermediate results.
Let us recall that the aim of this section is to establish some necessary and sufficient conditions for the solvability of the following nonlinear elliptic problem: where p, q ∈ (1, +∞) are fixed.We also assume that α j : Ê → Ê ( j = 1, 2) satisfy the following assumptions.  st ≤ Λ j (s) + Λ j (t). If s = α j (t) or t =α j (s), then we also have equality, that is, Λ j (α j (s)) = sα j (s) − Λ j (s), j = 1, 2.
We note that in [25], the statement of Lemma 4.4 assumed that Λ j , Λ j are N -functions in the sense of Definition 2.5. However, the conclusion of that result holds under the weaker hypotheses of Lemma 4.4.
Define the functional J 2 : 2 → [0, +∞] by with the effective domain Proof. It is routine to check that J 2 is convex and proper. This follows easily from the convexity of Λ j and the fact that Λ j (0) = 0. To show the lower semicontinuity on 2 , let U n = (u n , v n ) ∈ D(J 2 ) be such that U n → U := (u, v) in 2 and J 2 (U n ) ≤ C for some constant C > 0. Since U n → U in 2 , then there is a subsequence, which we also denote by U n = (u n , v n ), such that u n → u a.e. on Ω and v n → v σ -a.e. on Γ. Since Λ j (·) are continuous (thus, lower-semicontinuous), we have By Fatou's Lemma, we obtain Hence, J 2 is lower semicontinuous on 2 .
We have the following result whose proof is contained in [25, Chap. III, Section 3.1, Theorem 2]. Lemma 4.6. Let α j ( j = 1, 2) satisfy Assumption 4.3 and assume that there exist constants Let the operator B 2 be defined by We have the following result.
Noting that ∂ J 3,ρ is also a single-valued operator (which follows from the assumptions on α j and Λ j ), we easily obtain (4.10), and this completes the proof of the first part.
The following lemma is the main ingredient in the proof of Theorem 4.11 below.
Proof. We show that condition (4.14) is necessary. Let F := ( f , g) ∈ 2 and let U = u, u |∂ Ω ∈ D(B 3 ) ⊂ V ρ be a weak solution of B 3 U = F. Then, by definition, for every Hence, and so (4.14) holds. This completes the proof of part (a).
We show that the condition (4.15) is sufficient.
By definition, one has that C ⊂ R (B 2 ) since c 1 = α 1 (d 1 ) for some constant function d 1 on Ω and c 2 = α 2 (d 2 ) for some constant function d 2 on ∂ Ω. Let F ∈ 2 be such that (4.15) holds. We must show F ∈ R (B 3 ). By (4.15), we may choose C = (c 1 , c 2 ) ∈ C such that Then, for F ∈ 2 , we have F = F 1 + F 2 with Obviously, F 2 = C ∈ R (B 2 ). Hence, it is readily seen that (ii) Next, denote by Ê (x, r) the open ball in Ê of center x and radius r > 0. Since there exists δ > 0 such that the open ball Since the mapping F → F Ω from 2 into Ê is continuous, then there exists ε > 0 such that for all G ∈ 2 satisfying |F − G | 2 < ε. It finally follows from part (i) above that (R (B 1 ) + R (B 2 )) contains an ε-ball in 2 centered at F. Therefore, Consequently, problem (4.1) is (weakly) solvable for every function F = ( f , g) ∈ 2 , if (4.15) holds. This completes the proof of the theorem.

Remark 4.12.
It is important to remark that in order to prove Theorem 4.11, we do not require that (α 1 (u), α 2 (u)) should belong to 2 , for every U = (u, u |Γ ) ∈ D(J 3,ρ ). In particular, only the assumption (4.11) was needed. However, if this happens, then we get the much stronger result in (4.12) which would require that the nonlinearities α 1 , α 2 satisfy growth assumptions at infinity.
We conclude this section with the following corollary and some examples. Remark 4.14. Similar results to Theorem 4.11 and Corollary 4.13 were also obtained in [12,Theorem 4.4], but only when p = q = 2.

this problem has a weak solution if
which yields the classical Landesman-Lazer result (see (1.6)) for g ≡ 0 and p = 2.
has a weak solution if Let us now choose α (s) = arctan (s) in (4.16). Then, it is easy to check that is monotone increasing on Ê + and that it satisfies Λ(2t) ≤ C 2 Λ(t), ∀t ∈ Ê, for some constant C 2 > 1. Therefore, (4.17) becomes the necessary and sufficient condition (4.18)

A PRIORI ESTIMATES
Let Ω ⊂ Ê N be a bounded Lipschitz domain with boundary ∂ Ω. Recall that 1 < p, q < ∞, We consider the nonlinear elliptic boundary value problem formally given by where f ∈ L p 1 (Ω) and g ∈ L q 1 (∂ Ω) for some 1 ≤ p 1 , q 1 ≤ ∞. If ρ = 0, then the boundary conditions in (5.1) are of Robin type. Existence and regularity of weak solutions for this case have been obtained in [5] for p = 2 (see also [29] for the linear case) and for general p in [6]. Therefore, we will concentrate our attention to the case ρ = 1 only; in this case, the boundary condition in (5.1) is a generalized Wentzell-Robin boundary condition. For the sake of simplicity, from now on we will also take b ≡ 1.

Existence and uniqueness of weak solutions of perturbed equations. Let
Then for every 1 < p, q < ∞, V endowed with the norm is a reflexive Banach space. Recall that ρ = 1. Throughout the following, we denote by V ′ the dual of V .

Definition 5.3.
A function U = (u, u| ∂ Ω ) ∈ V is said to be a weak solution of (5.1), if for every V ∈ V = (v, v| ∂ Ω ), provided that the integrals on the right-hand side exist. Here, This shows A (U, ·) ∈ V ′ , for every U ∈ V . Next, let U,V ∈ V . Then, using (2.11) and the fact that α j (x, ·) are monotone nonde- This shows that A is monotone. The estimate (5.7) also shows that Thus, A is strictly monotone. The continuity of the norm function and the continuity of α j (x, ·), j = 1, 2 imply that A is hemicontinuous.
The following result is concerned with the existence and uniqueness of weak solutions to problem (5.1).
Proof. Let ·, · denote the duality between V and V ′ . Then, from Lemma 5.4, it follows that for each U ∈ V , there exists A(U) ∈ V ′ such that for every V ∈ V . Hence, this relation defines an operator A : V → V ′ , which is bounded by (5.6). Exploiting Lemma 5.4 once again, it is easy to see that A is monotone and coercive. It follows from Brodwer's theorem (see, e.g., [11,Theorem 5.3.22] Since W 1,p (Ω) ֒→ L p (Ω) and W 1,q (∂ Ω) ֒→ L q (∂ Ω) with dense injection, by duality, we have X p * ,q * (Ω, µ) ֒→ V ′ . Since Ω is bounded and σ (∂ Ω) < ∞, we obtain that This shows the existence of weak solutions. The uniqueness follows from the fact that A is strictly monotone (cf. Lemma 5.4). This completes the proof of the theorem. (a) Let 1 < p < N, 1 < q < p(N − 1)/N, p 1 ≥ p h and q 1 ≥ q k . Then for every ( f , g) ∈ X p 1 ,q 1 (Ω, µ), there exists a function U ∈ V which is the unique weak solution to (5.1). (b) Let 1 < q < N − 1, 1 < p < Nq/(N − 1), p 1 ≥ p h and q 1 ≥ q h . Then for every ( f , g) ∈ X p 1 ,q 1 (Ω, µ), there exists a function U ∈ V which is the unique weak solution to (5.1).
In order to prove the second part, we use the the embeddings W 1,p (Ω) ֒→ L p s (Ω), W 1,p (Ω) ֒→ L q s (∂ Ω) and proceed exactly as above. We omit the details.

Properties of the solution operator of the perturbed equation.
In the sequel, we establish some interesting properties of the solution operator A to problem (5.1). We begin by assuming the following.

This implies that
. Therefore, the operator A is injective and hence, A −1 exists. Since for every from the coercivity of A (see (5.8)), it is not difficult to see that for all n ∈ AE. Let U n := A −1 (F n ) and U = A −1 (F). Since {F n } is a bounded sequence and A −1 is bounded, we have that {U n } is bounded in V . Thus, we can select a subsequence, which we still denote by {U n } , which converges weakly to some function V ∈ V . Since Therefore, U n → V strongly in V . Since A is continuous and it follows from the injectivity of A, that U = V . This shows that which contradicts (5.14). Hence, A −1 : V ′ → V is continuous. The proof is finished. (a) If 2 ≤ p < N, 2 ≤ q < p(N − 1)/N, p 1 ≥ p h and q 1 ≥ q k , then A −1 : X p 1 ,q 1 (Ω, µ) → X p s ,q t (Ω, µ) is continuous and bounded. Moreover, A −1 : X p 1 ,q 1 (Ω, µ) → V ∩ X r,s (Ω, µ) is compact for every r ∈ (1, p s ) and s ∈ (1, q s ).
Proof. We only prove the first part. The second part of the proof follows by analogy and is left to the reader. Let 2 ≤ p < N, 2 ≤ q < p(N − 1)/N, p 1 ≥ p h and q 1 ≥ q k and let F ∈ X p 1 ,q 1 (Ω, µ). Proceeding exactly as in the proof of Theorem 5.8, we obtain Hence, the operator A −1 : X p 1 ,q 1 (Ω, µ) → X p s ,q t (Ω, µ) is bounded. Finally, using the facts that X p 1 ,q 1 (Ω, µ) ֒→ V ′ , A −1 : V ′ → V is continuous and V ֒→ X p s ,q t (Ω, µ), we easily deduce that A −1 : X p 1 ,q 1 (Ω, µ) → X p s ,q t (Ω, µ) is continuous. Now, let 1 < r < p s and 1 < s < q s . Since the injection V ֒→ X r,s (Ω, µ) is compact, then by duality, the injection X r ′ ,s ′ (Ω, µ) ֒→ (V ) * is compact for every r ′ > p ′ s = p h and s ′ > q ′ s = q h . This, together with the fact that A −1 : (V ) * → V is continuous and bounded, imply that A −1 : X p 1 ,q 1 (Ω, µ) → V is compact for every p 1 > p h and q 1 > q h .
It remains to show that A −1 is also compact as a map into X r,s (Ω, µ) for every r ∈ (1, p s ) and s ∈ (1, q s ). Since A −1 is bounded, we have to show that the image of every bounded set B ⊂ p 1 ,q 1 (Ω, µ) is relatively compact in X r,s (Ω, µ) for every r ∈ (1, p s ) and s ∈ (1, q s ). Let U n be a sequence in A −1 (B). Let F n = A(U n ) ∈ B. Since B is bounded, then the sequence F n is bounded. Since A −1 is compact as a map into V , it follows that there is a subsequence F n k such that A −1 (F n k ) → U ∈ V . We may assume that U n = A −1 (F n ) → U in V and hence, in X p,p (Ω, µ). It remains to show that U n → U in X r,s (Ω, µ). Let r ∈ [p, p s ) and s ∈ [p, q s ). Since U n := (u n , u n | ∂ Ω ) is bounded in X p s ,q s (Ω, µ), a standard interpolation inequality shows that there exists τ ∈ (0, 1) such that As U n converges in X p,p (Ω, µ), it follows from the preceding inequality that U n is a Cauchy sequence in X r,s (Ω, µ) and therefore converges in X r,s (Ω, µ). Hence, A −1 : X p 1 ,q 1 (Ω, µ) → V ∩ X r,s (Ω, µ) is compact for every r ∈ [p, p s ) and s ∈ [p, q s ). The case r, s ∈ (1, p) follows from the fact that X p,p (Ω, µ) ֒→ X r,s (Ω, µ) and the proof is finished 5.4. Statement and proof of the main result. We will now establish under what conditions the operator A −1 maps X p 1 ,q 1 (Ω, µ) boundedly and continuously into X ∞ (Ω, µ). The following is the main result of this section. (a) Suppose 2 ≤ p < N and 2 ≤ q < ∞. Let Let f ∈ L p 1 (Ω), g ∈ L q 1 (∂ Ω) and U,V ∈ V be such that for every function Φ = (ϕ, ϕ| ∂ Ω ) ∈ V , Then there is a constant C = C(N, p, q, Ω) > 0 such that Let f ∈ L p 1 (Ω), g ∈ L q 1 (∂ Ω) and U,V ∈ V satisfy (5.16). Then there is a constant C = C(N, p, q, Ω) > 0 such that ≤ C( f p 1 ,Ω + g q 1 ,∂ Ω ).