Research Article | Open Access
Ciprian G. Gal, Mahamadi Warma, "Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions", Advances in Mathematical Physics, vol. 2017, Article ID 5196513, 20 pages, 2017. https://doi.org/10.1155/2017/5196513
Nonlinear Elliptic Boundary Value Problems at Resonance with Nonlinear Wentzell Boundary Conditions
Given a bounded domain with a Lipschitz boundary and , we consider the quasilinear elliptic equation in complemented with the generalized Wentzell-Robin type boundary conditions of the form on . In the first part of the article, we give necessary and sufficient conditions in terms of the given functions , and the nonlinearities , , for the solvability of the above nonlinear elliptic boundary value problems with the nonlinear boundary conditions. In other words, we establish a sort of “nonlinear Fredholm alternative” for our problem which extends the corresponding Landesman and Lazer result for elliptic problems with linear homogeneous boundary conditions. In the second part, we give some additional results on existence and uniqueness and we study the regularity of the weak solutions for these classes of nonlinear problems. More precisely, we show some global a priori estimates for these weak solutions in an -setting.
Dedicated to the 70th birthday of Jerome A. Goldstein
Let , , be a bounded domain with a Lipschitz boundary and consider the following nonlinear boundary value problem with nonlinear second-order boundary conditions:where , , for some constant , is either or , and , are monotone nondecreasing functions such that . Moreover, is the -Laplace operator, , and , are given real-valued functions. Here, denotes the usual -dimensional Lebesgue measure in and denotes the restriction to of the -dimensional Hausdorff measure. Recall that coincides with the usual Lebesgue surface measure since has a Lipschitz boundary, and denotes the normal derivative of in direction of the outer normal vector . Furthermore, is defined as the generalized -Laplace-Beltrami operator on ; that is, , . In particular, and become the well-known Laplace and Laplace-Beltrami operators on and , respectively. Here, for any real-valued function ,where denotes the directional derivative of along the tangential directions at each point on the boundary, whereas denotes the tangential gradient at . It is worth mentioning again that when in (1), the boundary conditions are of lower order than the order of the -Laplace operator, while, for , we deal with boundary conditions which have the same differential order as the operator acting in the domain . Such boundary conditions arise in many applications, such as phase-transition phenomena (see, e.g., [1, 2] and the references therein), and have been studied by several authors (see, e.g., [3–7]).
In , the authors have formulated necessary and sufficient conditions for the solvability of (1) when , 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 , bythis result reads that a necessary condition for the existence of a weak solution of (1) is thatwhile a sufficient condition iswhere denotes the range of , , and denotes the interior of the set .
Relation (4) turns out to be both necessary and sufficient if either of the sets or is an open interval. This particular result was established in [4, 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:which is only a special case of (1) (i.e., , , and ). According to [4, Theorem 3] (see also (5)), this problem has a weak solution ifwhich yields the result of Landesman and Lazer  for . This last condition is both necessary and sufficient when the interval is open. This was put into an abstract context and significantly extended by Brézis and Haraux . Their work was much further extended by Brézis and Nirenberg . The goal of the present article is comparable to that of  since we want to establish similar conditions to (5) and (7) for the existence of solutions to (1) when , with main emphasis on the generality of the boundary conditions.
Theorem 1. Let be odd, monotone nondecreasing, continuous function such that . Assume that the functions satisfyfor some constants , . If is a weak solution of (1) (in the sense of Definition 30 below), thenConversely, ifthen (1) has a weak solution.
Our second main result of the paper deals with a modified version of (1) which is obtained by replacing the functions and in (1) by and , respectively, and also allowing , to depend on . Under additional assumptions on , and under higher integrability properties for the data , 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) with respect to the data can be also established. In particular, we prove the following.
Theorem 2. Let all the assumptions of Theorem 1 be satisfied for the functions , . Moreover, for each , assume that , as , and , as , respectively.(a)Then, for every with there exists a unique weak solution to problem (1) (in the sense of Definition 40 below) which is bounded.(b)Let , , be such that for some constants . Then, the weak (bounded) solution of problem (1) depends continuously on the data . Precisely, let us indicate by the unique solution corresponding to the data , for each . Then, the following estimate holds: for some nonnegative function , , which can be computed explicitly.
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) satisfies a sort of “quasilinear” version of the Fredholm alternative (cf. Theorem 20), which is needed in order to obtain the result in Theorem 1. Finally, in Sections 4 and 5, we provide detailed proofs of Theorems 1 and 2. We also illustrate the application of these results with some examples.
2. 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 [11–15]. We will also introduce some notations.
2.1. Maximal Monotone Operators
Let be a real Hilbert space with scalar product .
Definition 3. Let be a closed (nonlinear) operator. The operator is said to be(i)monotone if for all one has (ii)maximal monotone if it is monotone and the operator is invertible.
Next, let be a real reflexive Banach space which is densely and continuously embedded into the real Hilbert space , and let be its dual space such that .
Definition 4. Let be a continuous map.(a)The map is called a nonlinear form on if for all one has , that is, if is linear and bounded in the second variable.(b)The nonlinear form is said to be(i)monotone if ;(ii)hemicontinuous if , ;(iii)coercive, if .
Now, let be a proper, convex, lower semicontinuous functional with effective domain The subdifferential of the functional is defined by By a classical result of Minty  (see also [12, 14]), is a maximal monotone operator.
2.2. Functional Setup
Let be a bounded domain with a Lipschitz boundary . For , we let be the first-order Sobolev space; that is,Then , endowed with the norm is a Banach space, where we have set Since has a Lipschitz boundary, it is well-known that there exists a constant such thatwhere if and if . Moreover the trace operator initially defined for has an extension to a bounded linear operator from into where if and if . Hence, there is a constant such thatThroughout the remainder of this article, for , we let
For , we define the Sobolev space to be the completion of the space with respect to the norm where we recall that denotes the tangential gradient of the function at the boundary . It is also well-known that is continuously embedded into where if and if . Hence, for , there exists a constant such that
Let denote the -dimensional Lebesgue measure and let the measure on be defined for every measurable set by For , we define the Banach space endowed with the norm if , and If , we will simply denote .
Identifying each function with , we have that is a subspace of .
For , we endow with the norm while is endowed with the norm It follows from (21) and (22) that is continuously embedded into , with and given by (23), for . Moreover, by (21) and (26), is continuously embedded into .
2.3. 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 5. Let be a complete measure space. We call a function a Musielak-Orlicz function on if(a) is nontrivial, even, and convex for , a.e. ;(b) is vanishing and continuous at for , a.e. ;(c) is left continuous on ;(d) is -measurable for all ;(e).
The complementary Musielak-Orlicz function is defined by It follows directly from the definition that for (and hence for all )
Definition 6. We say that a Musielak-Orlicz function satisfies the -condition if there exists a set of -measure zero and a constant such that for all and every .
We say that satisfies the -condition if there is a set of -measure zero and a constant such that for all and all .
Definition 7. A function is called an -function if
(i) is even, strictly increasing, and convex;(ii) if and only if ;(iii) and .We say that an -function satisfies the -condition if there exists a constant such that and it satisfies the -condition if there is a constant such that For more details on -functions, we refer to the monograph of Adams [18, Chapter VIII] (see also [19, Chapter I], [20, Chapter I]).
Remark 8. For an -function , we let be its left-sided derivative. Then is left continuous on and nondecreasing. Let be given by Then As before for all Moreover, if or then we have equality; that is,The function is called the complementary -function of . It is also known that an -function satisfies the -condition if and only iffor some constant and for all , where is the left-sided derivative of .
Lemma 9. Let be an -function which satisfies the -condition with the constant and let be its complementary -function. Then satisfies the -condition with the constant .
Proof. We have Since for all and and are decreasing, we get, for , that Now let . Then for Hence, .
Corollary 10. Let be a Musielak-Orlicz function such that is an -function for , a.e. on . If satisfies the -condition, then satisfies the -condition.
Definition 11. Let be a Musielak-Orlicz function. Then the Musielak-Orlicz space associated with is defined by where On this space we consider the Luxemburg norm defined by
Proposition 12. Let be a Musielak-Orlicz function which satisfies the -condition. Then
Proof. If satisfies the -condition, then there exists a set of measure zero such that for every there exists for all and all . Let be fixed. For there exists satisfying the above inequality. We will show that whenever . Assume that and let be such that . Then for all . If we assume that the last inequality does not hold, then and this clearly contradicts the definition of . Therefore, we must haveFrom (53), (56), we obtain The proof is finished.
Corollary 13. Let be a Musielak-Orlicz function such that is an -function for , a.e. on . If its complementary -function satisfies the -condition, then satisfies the -condition and
2.4. Some Tools
For the reader’s convenience, we report here below some useful inequalities which will be needed in the course of investigation.
Lemma 14. Let and . Then, there exists a constant such thatIf , then there exists a constant such that
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.
Lemma 15. Let be a nonnegative, nonincreasing function such that there are positive constants and () such that Then with .
3. The Fredholm Alternative
In what follows, we assume that is a bounded domain with Lipschitz boundary . Let satisfy for some constant . Let be the real Hilbert space . Then, it is clear that is isomorphic to with equivalent norms.
Next, let and be fixed. We define the functional by settingwhere the effective domain is given by .
Throughout the remainder of this section, we let . The following result can be obtained easily.
Proposition 16. The functional defined by (62) is proper, convex, and lower semicontinuous on .
The following result contains a computation of the subdifferential for the functional .
Remark 17. Let and let . Then, by definition, and, for all , we have Let , , and set above. Dividing by and taking the limit as , we obtain thatwhere we recall that Choosing with (the space of test functions) and integrating by parts in (64), we obtain Therefore, the single-valued operator is given by
Since the functional is proper, convex, and lower semicontinuous, it follows that its subdifferential is a maximal monotone operator.
In the following two lemmas, we establish a relation between the null space of the operator and its range.
Lemma 18. Let denote the null space of the operator . Then that is, consists of all the real constant functions on .
Proof. We say that if and only if (by definition) is a weak solution ofA function is said to be a weak solution of (70), if, for every , there holdsLet with . Then it is clear that .
Conversely, let . Then, it follows from (71) that Since is bounded and connected, this implies that is equal to a constant. Therefore, and this completes the proof.
Lemma 19. The range of the operator is given by
Proof. Let . Then there exists such that . More precisely, for every , we have