Abstract
We provide the existence of a positive solution for the quasilinear elliptic equation
in Ω under the Dirichlet boundary condition. As a special case (
MSC: 35J92, 35P30.
Keywords:
nonhomogeneous elliptic operator; positive solution; the first eigenvalue with weight; approximation1 Introduction
In this paper, we consider the existence of a positive solution for the following quasilinear elliptic equation:
where
Throughout this paper, we assume that the mapAand the nonlinear termfsatisfy the following assumptions (A) and (f), respectively.
(A)
(i)
(ii)
(iii)
(iv)
(v)
(f) f is a continuous function on
for every
In this paper, we say that
for all
A similar hypothesis to (A) is considered in the study of quasilinear elliptic problems
(see [[1], Example 2.2.], [25] and also refer to [6,7] for the generalized pLaplace operators). From now on, we assume that
In particular, for
In the case where f does not depend on the gradient of u, there are many existence results because our equation has the variational structure
(cf.[1,4,8]). Although there are a few results for our equation (P) with f including ∇u, we can refer to [7,9] and [10] for the existence of a positive solution in the case of the
In [9], the positivity of a solution is proved by the comparison principle. However, since we are not able to do it for our operator in general, after we provide a nonnegative and nontrivial solution as a limit of positive approximate solutions (in Section 2), we obtain the positivity of it due to the strong maximum principle for our operator.
1.1 Statements
To state our first result, we define a positive constant
which is equal to 1 in the case of
(f1) There exist
where
Theorem 1Assume (f1). Then equation (P) has a positive solution
andνdenotes the outward unit normal vector on∂Ω.
Next, we consider the case where A is asymptotically
(AH0) There exist a positive function
Under (AH0), we can replace the hypothesis (f1) with the following (f2):
(f2) There exist
Theorem 2Assume (AH0) and (f2). Then equation (P) has a positive solution
Throughout this paper, we may assume that
1.2 Properties of the map A
Remark 3 The following assertions hold under condition (A):
(i) for all
(ii)
(iii)
where
Proposition 4 ([[3], Proposition 1])
Let
for
2 Constructing approximate solutions
Choose a function
In [7], the case
Lemma 5Suppose (f1) or (f2). Then there exists
Proof From the growth condition of
holds, where
Proposition 6If
Proof Set
If
due to Remark 3(iii) and
where
provided
Lemma 7Suppose (f1) or (f2). If
Proof Taking
because of
The following result can be shown by the same argument as in [[9], Theorem 3.1].
Proposition 8Suppose (f1) or (f2). Then, for every
Proof Fix any
for u,
for every
where
Because (11) with
as
Finally, we shall prove that
Since l is arbitrary, (12) holds for every
3 Proof of theorems
Lemma 9Let
holds, where
Proof Because of
in Ω by (ii) and (iii) in Remark 3 and Young’s inequality. □
Lemma 10Assume that
holds.
Proof First, we note that
in Ω. Similarly, we also have
The conclusion follows from (15) and (16). □
Under (f1) or (f2), we denote a solution
Lemma 11Assume (f1) or (f2). Let
Proof Taking
by Remark 3(iii), the growth condition of f, Hölder’s inequality and the continuity of the embedding of
Lemma 12Assume (f1) or (f2). Then
Proof Fix any
by Lemma 5 and
Therefore, (17) and (18) imply the inequality
Lemma 13Assume (f2) and (AH0). Let
holds, where
Proof Note that
because of
3.1 Proof of main results
Proof of Theorems
Let
Proof of Theorem 1 Let
This is a contradiction. □
Proof of Theorem 2 Since
as
As a result, by taking a limit superior with respect to n in (19), we have
Competing interests
The author declares that she has no competing interests.
Acknowledgements
The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks Professor Dumitru Motreanu for giving her the opportunity of this work. The author thanks referees for their helpful comments.
References

Motreanu, D, Papageorgiou, NS: Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator. Proc. Am. Math. Soc.. 139, 3527–3535 (2011). Publisher Full Text

Damascelli, L: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. Henri Poincaré. 15, 493–516 (1998)

Motreanu, D, Motreanu, VV, Papageorgiou, NS: Multiple constant sign and nodal solutions for nonlinear Neumann eigenvalue problems. Ann. Sc. Norm. Super. Pisa, Cl. Sci.. 10, 729–755 (2011)

Miyajima, S, Motreanu, D, Tanaka, M: Multiple existence results of solutions for the Neumann problems via super and subsolutions. J. Funct. Anal.. 262, 1921–1953 (2012). Publisher Full Text

Motreanu, D, Tanaka, M: Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application. Pac. J. Math.. 265(1), 151–184 (2013)

Kim, YH: A global bifurcation for nonlinear equations with nonhomogeneous part. Nonlinear Anal.. 71, 738–743 (2009). Publisher Full Text

Ruiz, D: A priori estimates and existence of positive solutions for strongly nonlinear problems. J. Differ. Equ.. 2004, 96–114 (2004)

Tanaka, M: Existence of the Fučík type spectrums for the generalized pLaplace operators. Nonlinear Anal.. 75, 3407–3435 (2012). Publisher Full Text

Faria, L, Miyagaki, O, Motreanu, D: Comparison and positive solutions for problems with (P,Q)Laplacian and convection term. Proc. Edinb. Math. Soc. (to appear)

Zou, HH: A priori estimates and existence for quasilinear elliptic equations. Calc. Var.. 33, 417–437 (2008). Publisher Full Text

Lieberman, GM: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal.. 12, 1203–1219 (1988). Publisher Full Text

Fučík, S, John, O, Nečas, J: On the existence of Schauder bases in Sobolev spaces. Comment. Math. Univ. Carol.. 13, 163–175 (1972)

Deimling, K: Nonlinear Functional Analysis, Springer, New York (1985)

Anane, A: Etude des valeurs propres et de la résonnance pour l’opérateur plaplacien. C. R. Math. Acad. Sci. Paris. 305, 725–728 (1987)

Cuesta, M: Eigenvalue problems for the pLaplacian with indefinite weights. Electron. J. Differ. Equ.. 2001(33), 1–9 (2001)