Abstract
The paper presents existence results for nonlinear elliptic problems under a nonhomogeneous Dirichlet boundary condition. The considered elliptic equations exhibit nonlinearities containing derivatives of the solution.
MSC: 35H30, 35A16.
Keywords:
quasilinear elliptic problem; nonhomogeneous Dirichlet boundary conditions; existence result1 Introduction
The aim of the paper is twofold: first, to study nonlinear elliptic problems under nonhomogeneous Dirichlet boundary condition; second, to incorporate in the problem statement nonlinearities exhibiting derivatives of the solution. These requirements need to develop a nonstandard approach, in particular prevent the use of variational methods.
Specifically, we study two problems on a bounded domain
where
Definition 1 A (weak) solution of problem (1) is an element
Next we focus on nonhomogeneous Dirichlet problems where, contrary to problem (1), the dependence with respect to the gradient ∇u of the solution u is not expressed in a divergence form, namely
Here
Definition 2 A (weak) solution of problem (2) is an element
for all
Problems of type (1) and (2) have been investigated in settings that are different
from ours (see, e.g., [16]). For instance, problem (1) is studied in [3] when
Our results are stated as Theorems 1 and 2. They are existence and location theorems
on problems (1) and (2), respectively, guaranteeing solutions in the sense of Definitions
1 and 2 that fulfill an estimate
The rest of the paper is organized as follows. Section 2 is devoted to problem (1). Section 3 studies problem (2).
2 Result on problem (1)
Throughout the paper the notation
Let
We suppose the following hypotheses on the data a, b, f, and g in problem (1):
(H_{1}) There is a Carathéodory function
(H_{2}) There are constants
(H_{3}) The functions a, f are bounded on the set
(H_{4}) There is a Carathéodory function
and
Remark 1 The constants
Remark 2 Due to their different structure and requirements, the two terms in (1) that are in divergence form cannot be combined.
Remark 3 The last part of hypothesis (H_{4}) incorporates the monotonicity condition
as well as the Lipschitz condition
and it is more general than both of them.
The result that we set forth in this section is the following theorem ensuring existence and location of solution for problem (1).
Theorem 1Assume that hypotheses (H_{1})(H_{4}) are satisfied. Then problem (1) has at least one solution
with
Proof Consider the set
which is a nonempty, bounded, closed, convex subset in
Claim 1: Given
Note that Claim 1 is equivalent to solving uniquely the problem
Here
and
for all
With the fixed element
From hypotheses (H_{1}), (H_{3}), (H_{4}), and because
and
Estimate (5) guarantees that the operator A is bounded (in the sense to be bounded on bounded sets). It is easily seen that (6) implies that A is coercive, that is,
Moreover, relations (5)(6) ensure that the operator A is maximal monotone, so pseudomonotone (see, e.g., [[7], §2.3.1]). Since A is bounded, coercive, and pseudomonotone, it is surjective (see, e.g., [[7], Theorem 2.99]), whence the existence of
Now, taking advantage of Claim 1, we define the operator
Claim 2: The mapping
Let
Combining this formula with (H_{1}), (H_{3}), (H_{4}), (3) and the CauchySchwarz inequality, we obtain
Taking into account hypothesis (H_{4}) leads to
Set
We claim that
To this end, we show that any subsequence of
Similarly, we have
Then, in view of (7), we infer that
With the truncation function
consider the operator
Note that S takes values in
Claim 3: The mapping
Since
Let
Then, as in the proof of Claim 2, from assumptions (H_{1}) and (H_{4}) we obtain that
whence, by (H_{4}),
with a constant
Using (11), we derive
with constants
Claim 4: Let
The existence of a point
which reads as
By the assumption that
whence
By Claims 3 and 4, the operator T admits a fixed point
3 Result on problem (2)
The hypotheses on the data a,
(
(
(
Remark 4 As in the case of problem (1), we note that the constant functions
Now we state our result of existence and location of solutions for problem (2).
Theorem 2Assume that (H_{1}), (
with
Proof We follow the pattern of proof of Theorem 1. Hence, using the constants
which is a nonempty, bounded, closed, convex subset of
Claim 1: For every
As in the proof of Theorem 1, first we note that Claim 1 is equivalent to proving that the problem
admits a unique solution, where
and
For
for all
which ensures that
Since
On the basis of the reasoning in (14), the following estimate holds
for all
where
As in the proof of Theorem 1, we introduce the operator
Claim 2: The mapping
In order to prove this assertion, we proceed as in the proof of Claim 2 in Theorem 1.
Fix
A straightforward calculation entails
Combining (H_{1}), (
with μ and
Proceeding as in (8) we show that
Now it suffices to combine (18), (19), (20) and recall that
Following the approach developed in the proof of Theorem 1, we introduce the operator
Claim 3: The mapping
Claim 2 readily implies that the mapping
where
with a constant
with
Claim 4: If
Let
Testing in
In
Then hypothesis (
Combining with (22), (H_{1}), and (
It turns out that
Now we can conclude the proof. Claims 3 and 4 ensure that there exists a fixed point
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DM and VVM jointly worked and obtained all the results presented in the paper and participated equally in the preparation of the paper. Both authors read and approved the final manuscript.
Acknowledgements
The second author is supported by the Marie Curie IntraEuropean Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement No. PIEFGA2010274519).
References

Bensoussan, A, Boccardo, L, Murat, F: On a nonlinear partial differential equation having natural growth terms and unbounded solution. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 5, 347–364 (1988)

Boccardo, L, Giachetti, D, Diaz, JI, Murat, F: Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms. J. Differ. Equ.. 106, 215–237 (1993). Publisher Full Text

Buscaglia, G, Ciuperca, I, Jai, M: Existence and uniqueness for several nonlinear elliptic problems arising in lubrication theory. J. Differ. Equ.. 218, 187–215 (2005). Publisher Full Text

Ferone, V, Murat, F: Nonlinear problems having natural growth in the gradient: an existence result when the source terms are small. Nonlinear Anal.. 42, 1309–1326 (2000). Publisher Full Text

Gilbarg, D, Trudinger, NS: Elliptic Partial Differential Equations of Second Order, Springer, Berlin (2001)

Ladyzhenskaya, OA, Ural’tseva, NN: Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968)

Carl, S, Le, VK, Motreanu, D: Nonsmooth Variational Problems and Their Inequalities. Comparison Principles and Applications, Springer, New York (2007)

Zeidler, E: Nonlinear Functional Analysis and Its Applications. I. FixedPoint Theorems, Springer, New York (1986)