Keywords:elliptic mixed boundary value problem; positive solutions; mountain pass theorem; Sobolev embedding theorem
1 Introduction and preliminaries
This paper is concerned with the existence of positive solutions of the following elliptic mixed boundary value problem:
The eigenvalue problem of (1) is studied by Liu and Su in 
There have been many papers concerned with similar problems at resonance under the boundary condition; see [2-10]. Moreover, some multiplicity theorems are obtained by the topological degree technique and variational methods; interested readers can see [11-17]. Problem (1) is different from the classical ones, such as those with Dirichlet, Neuman, Robin, No-flux, or Steklov boundary conditions.
In this paper, we assume is a closed subspace of . We define the norm in V as , is the norm, is the norm, is the trace operator with for all , that is continuous and compact (see ). Furthermore, we define , for (see ). Then, by (S3), we obtain
where , , . Moreover, by (S1) and the Strong maximum principle, a nonzero critical point of J is in fact a positive solution of (1). In order to find critical points of the functional (6), one often requires the technique condition, that is, for some , , ,
It is easy to see that the condition (AR) implies that , that is, must be superlinear with respect to u at infinity. In the present paper, motivated by  and , we study the existence and nonexistence of positive solutions for problem (1) with the asymptotic behavior assumptions (S3) of f at zero and infinity. Moreover, we also study superlinear of f at infinity with in (S3), which is weaker than the (AR) condition, that is the (AR) condition does not hold.
In order to get our conclusion, we define the minimization problem
Theorem 1Let conditions (S1) to (S3) hold, then:
2 Some lemmas
We need the following lemmas.
Proof By the Sobolev embedding function and Fatou’s lemma, it is easy to know that and there exists , which satisfies Λ, that is, . Furthermore, we assume , then could replace by . By the Strong maximum principle, we know a.e. in V. □
By (4) and (5), we obtain
Then (S2) implies that
On the other hand, by (8), one has
Lemma 5 (see )
3 Proofs of main results
(14) implies that
If is bounded in V, when Ω is bounded and , are subcritical, we can get has a subsequence strong convergence to a critical value of J, and our proof is complete. So, to prove the theorem, we only need show that is bounded in V. Supposing that is unbounded, that is, as . We order
By (16), we have
Proof of Theorem 3 When , we can replace by in (11) and define c as in (12), then following the same procedures as in the proof of Theorem 1(ii), we need to show only that is bounded in V. For this purpose, let be defined as in (16). If is bounded in V, we know is a strong convergence in , is convergence a.e. , is a weak convergence in V, and .
In this section, we give two examples on : One satisfies (S1) to (S3) with , but does not satisfy the (AR) condition; the other illustrates how the assumptions on the boundary are not trivial and compatible with the inner assumptions in Ω.
Example 1 Set:
Example 2 Consider the following problem:
The author declares that he has no competing interests.
Li G carried out all studies in this article.
The author would like to thank the referees for carefully reading this article and making valuable comments and suggestions.
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