An existence result for anisotropic quasilinear problems
Introduction
In this work we study the degenerate (or singular) boundary value problem (BVP) where , and is a smooth bounded domain.
In order to formulate our assumptions on the function , we consider first small so that in the set the distance function satisfies that given any , there exists a unique such that (see also Section 2.3). We consider also a positive function and a fixed positive number such that
In particular, .
The assumptions for the function are the following:
(A1) and for any open set with , ;
(A2) for every ,
Assumptions (A1) and (A2) imply that the differential operator in (1.1) is uniformly elliptic on any subdomain of , but not necessarily near the boundary of the domain.
These hypotheses also yield that and (see Section 2.1), which will allow us to put Eq. (1.1) into an appropriate functional analytic setting.
To formulate the assumptions on the function , we first let be as in (1.3) and denote
Using that , we find that
Next, fix and set , if . Consider also a measurable function satisfying the following:
(B1) ;
(B2) there exist constants such that
In particular, and a.e. in .
As for the function , we assume it is a Caratheodory function satisfying the growth condition:
for every , there exists such that for every with and for a.e. ,
We use weighted Sobolev spaces to study the BVP (1.1). The Sobolev space is defined as the class of functions such that
The Sobolev space is defined as the closure of with respect to the norm .
A function is called a solution of (1.1) if and
A function is called a subsolution (supersolution respectively) of (1.1) if , for every with a.e. in , and .
Our first main result reads as follows.
Theorem 1.1 Let (A1)-(A2) and (B1)-(B2) hold and let satisfy (1.6). Assume that is a subsolution and is a supersolution to (1.1) such that in . Then the BVP (1.1) has a solution such that in .
The proof of Theorem 1.1 is motivated by the techniques described in Chapter 5 in [1]. As an application of Theorem 1.1, we present existence of a positive solution for the BVP where is a positive parameter and is a continuous function satisfying the following hypotheses:
(f1) is -sublinear, i.e., ;
(f2) there exists such that .
(f3) and .
Problem (1.9) finds its applications, for instance, in resource management models, see [2], and design of suspension bridges, see [3].
We present the function where , as an example of a nonlinearity satisfying conditions (f1), (f2) and (f3).
Condition in (f3) implies that is a supersolution to (1.9). Under this condition, finding a positive subsolution for (1.9) becomes a subtle matter. We refer the reader to [4], [5], [6], [7], [8] and references therein, for results concerned with related boundary value problems in the uniformly elliptic case.
The authors in [9] showed the existence of a positive solution of Eq. (1.9) either in a ball or in , under a similar set of assumptions on and , but assuming in addition the global radial symmetry of the coefficient and instead of (f3), assuming that there exist and such that
As remarked in [9], our hypotheses do not allow, in general, to obtain either -regularity of weak solutions to (1.9), see [10], or Hopf’s boundary lemma, see [11]. Instead, regularity of solutions relies on the local Hölder continuity due to Serrin, see [12]. In particular, global a priori -estimates and therefore the construction of a subsolution of (1.9) is a much more challenging task. In this work, tools from differential geometry help us overcome this difficulty.
More precisely, we say that has non-negative mean curvature if for every , where are the principal curvatures of .
This geometric notion generalizes convexity and it is equivalent to the monotonicity of the surface area element of , see [13]. This monotonicity states that the -dimensional volume of every subdomain does not decrease as approaches .
From the PDE point of view, condition (1.10) implies that the distance function is subharmonic in any subdomain of (see (1.2)) and in essence this fact will allow us to construct an appropriate subsolution to (1.9).
Our second main result reads as follows.
Theorem 1.2 Let be a smooth bounded domain such that has non-negative mean curvature. Let (A1)-(A2) and (B1)-(B2) hold and let satisfy (f1)-(f3). Then, there exists such that for every the BVP (1.9) has a solution which is positive in .
We refer the reader to [14], [15] for similar results in the uniformly elliptic setting.
In the case and , under slightly more restrictive assumptions on and with no assumptions on the geometry of , the authors in [16] proved uniqueness of positive solutions of (1.9) for large. We also refer the reader to [17] for non-existence of positive radial solutions of a related BVP in the superlinear setting.
We have considered assumption (A2) for the sake of clarity in our presentation. Although, we remark that Theorem 1.1, Theorem 1.2 are still valid if instead of (A2), we assume the existence of functions , both satisfying (1.3) and such that for every . The proofs require only straight forward changes and details are left to the reader.
The paper is organized as follows. In Section 2 we present the preliminary results and a priori estimates needed for the proof of Theorem 1.1. The proof of Theorem 1.1 is presented in Section 3. In Section 4 we provide the proof Theorem 1.2.
Section snippets
Weighted Sobolev spaces
Let , be a smooth bounded domain and let satisfy (A1) and (A2).
From these hypotheses and with the help of Fermi coordinates, we verify in Section 2.3 that and . Observe also that the Hölder inequality and the fact that yield .
Thus, the spaces and , defined in Section 1, are separable. They are also uniformly convex and therefore reflexive Banach spaces (see Chapter 1 in [18]).
We also notice from (A1) that for any with
Proof of Theorem 1.1
In this part we follow the scheme from Chapter 5 in [1] to prove Theorem 1.1. Also, we make use of the notations and conventions introduced in Sections 1 Introduction, 2 Preliminaries.
Assume hypotheses (A1)–(A2) and (B1)–(B2). Using (B2), let be such that where is as in (2.24).
Consider the space endowed with the norm and its subspace
Next, consider the space where
Proof of Theorem 1.2
In this final section we prove Theorem 1.2. Besides (A1)-(A2) and (B1)-(B2), we assume also (f1)-(f3). We adopt the notations and conventions from the previous sections.
Proposition 4.1 For every , there exist a supersolution to (1.9) with in .
Although the proof of Proposition 4.1 is rather standard, we include it here for the sake of completeness.
Proof Let be arbitrary. Set , where is the solution to (3.2) with and is a parameter to be chosen later. Since ,
Acknowledgment
The first author was supported by the Project LO1506 of the Ministry of Education, Youth and Sports of the Czech Republic .
The authors want to thank the referee for careful reading of the manuscript and the suggestions which improved the exposition of the text.
References (24)
- et al.
Existence and uniqueness for a class of quasilinear elliptic boundary value problems
J. Differential Equations
(2003) - et al.
Positive solutions for elliptic equations involving nonlinearities with falling zeroes
Appl. Math. Lett.
(2009) - et al.
Anisotropic semipositone quasilinear problems
J. Math. Anal. Appl.
(2017) Regularity for a more general class of quasilinear elliptic equations
J. Differential Equations
(1984)- et al.
Singular quasilinear elliptic problems on unbounded domains
Nonlinear Anal.
(2014) - et al.
- et al.
Nonlinear eigenvalue problems with semipositone structure
Electron. J. Diff. Eqns. Conf.
(2000) Nonlinearity in oscillating bridges
Electron. J. Differential Equations
(2013)- et al.
Uniqueness and stability of nonnegative solutions for semipositone problems in a ball
Proc. Amer. Math. Soc.
(1993) - et al.
Positive solutions for some semi-positone problems via bifurcation theory
Differential Integral Equations
(1994)