1. Introduction and Main Results
Consider the Navier boundary value problem involving the polyharmonic:
where , W is an open bounded domain in ( ) with smooth boundary , is continuous on . We look for the weak solutions of (1.1) which are the same as the critical points of the functional defined by
where , and
is denoted by and its dual spaces by . The Hilbert space is equipped with the standard scalar product and norm, respectively, given by:
is of with derivatives given by
Motivated by the fact that such kinds of problems (1.1) are often used to describe an important class of physical problems, many authors have widely developed various methods and techniques, such as variational method, critical point theory, lower and upper solutions, and Brower degree for looking for multiple solutions of elliptic equations involving biharmonic, p-biharmonic and polyharmonic type operators (see, for example  -  and the references therein). In  , the author considered the existence of positive solutions in semilinear critical problems for polyharmonic operators and proved the existence result in some general domain under the appropriate assumptions by topological methods. In  , by variational methods, they obtain the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator. By using Nehari manifold, Y.Y. Shang and L. Wang  considered (1.1) with critical growth and sublinear perturbation and obtained the existence of multiple nontrivial solutions. Moreover in  , a concrete example of application of such mathematical model to describe a physical phenomenon is also pointed out. By three critical points theorem obtained by B. Ricceri, Li and Tang  have obtained the existence of at least three weak solutions for a class of Navier boundary value problem involving the p-biharmonic. In  , combining the mountain pass theorem together with fountain theorem and with local linking theorem and symmetric mountain pass theorem, Li and Tang establish the existence of at least one solution and infinitely many solutions for a class of p(x)-biharmonic equations with Navier boundary condition, respectively. By using critical point theory, the authors  establish the existence of infinitely many weak solutions for a class of elliptic Navier boundary value problems depending on two parameters and involving the p-biharmonic operator. In  , a Navier boundary value problem is treated where the left-hand side of the equation involves an operator that is more general than the p-biharmonic. Further, by using the abstract and technical approach developed in    the authors are interested in looking for the existence of infinitely many weak solutions of perturbed p-biharmonic equations. For more results about (1.1) and its variants, we refer the interested readers to  -  and the references therein.
The aim of the manuscript is to consider the problem in a different case: based on the mountain pass theorem; we show that the Navier boundary value problem has at least a weak nontrivial solution for all . We now assume the nonlinearity satisfies the following conditions.
(F1) There exist constants such that
where is the Sobolev critical exponent.
The main theorem of this paper reads as follows.
Theorem 1. Suppose that (F1)-(F4) hold. Then Equation (1.1) has a weak nontrivial solution, for all .
This paper is organized as follows. In Section 2, we give the proof of the Theorem 1. In the sequel, the letter C or will be used to denote various positive constants whose exact value is irrelevant.
2. Proof of Theorem 1
Proof of Theorem 1. We will divide the proof of the theorem in three steps.
First step: the (P.S.) condition. We only prove Theorem 1 by taking . The case of a general will follow immediately. In fact, if and , we only let . Then (1.1) becomes
The nonlinear term g also satisfies condition (F1)-(F4), then the same conclusion as in the case holds. We denote I1 as I for convenience.
Let be any sequence in such that is bounded and converges to zero, that is,
Applying the Riesz-Fréchet representation theorem, there is such that
Thus, we obtain
In (2.1), taking , we obtain
From Hölder’s inequality, we get:
On the other hand, since
then (F1) together with , we have
Since is bounded and , by (2.2) and (2.4), we conclude that is bounded.
By (F1), for any there exists such that
Since is bounded in and is continuous on , then there exists a subsequence of , still denoted by and such that
Moreover there is an such that
Combining (2.6) and (2.7) with the Egorov theorem, there exists such that and converge to 0, and further
Also, since is bounded in , we derive that
On one hand, we have
Combining (2.5) and (2.9) with the fact , one has
By (2.8), there exists an integer such that for we have
Taking account of (2.10), for , we have
This proves that in .
Second step: mountain-pass geometric structure.
I has a mountain pass geometry; i.e., there exists and constants such that and
Indeed, using the assumption (F3), we have and such that
where denotes the lowest eigenvalue of with the Navier boundary condition. As shown in  , is the principal eigenvalue of
and there is a corresponding eigenfunction in W.
Hence, by (2.14), we have
This inequality together with the assumption (F2) implies that
with some . Since denotes the lowest eigenvalue of with the Navier boundary condition, it follows that for . Then the following estimates hold:
This implies that there exist constants r and r such that
From (F4) follows that, for all there exists , such that
By (3.4), we have
where is the principal eigenfunction of (2.15) corresponding to the principal eigenvalue . We fix so large that and . Let and then constants such that and satisfies (2.13), i.e. I has a mountain pass geometry.
Last step: critical value of I.
For in second step, we define
From Proposition 1, I satisfies the (P.S.) condition. By the Mountain Pass Theorem implies that is a critical value of I.
This work is supported by Natural Science Foundation of China (No. 11671331); Natural Science Foundation of Fujian Province (No. 2015J01585) and Scientific Research Foundation of Jimei University.