Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities

Abstract

This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, and where is the critical Sobolev exponent, . Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when be a bounded, smoothandstrictlystar-shapeddomain, .

1. Introduction and Main Results

The main purpose of this article is to investigate the existence and nonexistence of nontrivial solutions of the following problem: where is the exterior normal vector to , is a smooth-bounded domain in , , with is the critical Sobolev exponent, , . is the operator of the fourth order, so-called the p-biharmonic (or p-bilaplacian) operator. For , the linear operator is the iterated Laplacian that to a multiplicative positive constant appears often in the equations of Navier-Stokes as being a viscosity coefficient, and its reciprocal operator noted is the celebrated Green’s operator see [1].

The fourth-order differential equations with nonlinearity arise in the study of deflections of elastic beams on nonlinear elastic foundations. It furnishes a model to study travelling waves in suspension bridges. Lazer and McKenna [2] gave a survey of results in this direction. This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second-order problems which have been studied by many authors (see [3] and references therein). Thus, they become very significant in engineering and physics and many authors considered this type of equation in recent years, and we refer to [4–11] and the references therein. For this reason, the existence of solutions of p-biharmonic equations has been studied by several authors; see [12–15]. Li [16] establish the existence of at least two distinct weak solutions for the following singular elliptic problems involving a p-biharmonic operator, subject to Navier boundary conditions in a smooth-bounded domain in . with being a Carathéodory function. Wang [17] studied the existence and multiplicity to p-biharmonic equation with the Sobolev–Hardy term under the Dirichlet boundary conditions and the Navier boundary conditions, respectively.

Before giving our main results, we state here some definitions, notations, and known results.

1.1. Notations

Set a Banach space, with the norm for

We consider the following approximation equation: for any such that . The energy functional of (4) is defined by for all

We note that is a function on

A point is a weak solution of the Equation (1) if it satisfies where with .

Here, denotes the product in the duality .

In our work, we research the critical points as the minimizers of the energy functional associated to problem (1).

Let

From [18], is achieved.

Let with

Now, we can state our main results.

Theorem 1. Assume that , with is the critical Sobolev exponent, , and ; then, problem (1) has at least a nontrivial solution in .

Theorem 2. Let be a bounded, smooth, and strictly star-shaped domain, and with Then, problem (1) has no positive solutions in

This paper is organized as follows. In Section 2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2.

2. Some Preliminary Results

Definition 3 [1]. An operator : is hemicontinuous if is continuous from to , for all

Definition 4 [1]. An operator : is “ calculus of variations,” if it is bounded and if it can be represented by
where is an operator satisfying the following properties.
For all: ; is bounded, hemicontinuous of .
If converges weakly to in and if then, for all , the sequence definite by converges to in
If converges weakly to in , and if converges weakly to in , then

3. Existence Results

The energy functional associated to Equation (4) is defined on by

To prove Theorem 1, we use the three following lemmas.

Set with and .

Firstly, we need the following Lemmas.

Lemma 5. is.

Proof. Let such that . If and by the Hölder inequality, we obtain with
where It is enough to show that and are on
According to an algebraic relation of Li and Tang [3] and the Hölder inequality, one has where
Therefore is on and is , since is subcritical and by (16). Thus, is on

Lemma 6. satisfies the Palais-Smale condition.

Proof. Let be a Palais-Smale sequence in (i.e., is bounded and ). is bounded in . In fact, Combining (19), (20), and (16), we obtain that is bounded when and
The embedding is compact for ; then, there exists a subsequence, still denoted by , which converges in . Show that converges in
Set Then, we can write in the form In addition, verifies an algebraic relation [1], from where thus, and therefore Then, converges strongly in

Lemma 7. Suppose , with is the critical Sobolev exponent, , and ; then, there exist and positive constants such that (i)We have (ii)There exists when , with , such that .

Proof. (i)We haveThen, we get when small. (ii) and for an element ; then,Indeed
Let be such that However we obtain Owing to the fact that is continuous, then there exists such that

Proof of Theorem 8. From Lemmas 5–7, we deduce that there exists such that

Finally, for every problem (4) has solution such that . Thus, there exist with as Then, we get .

4. Nonexistence Result

By a Pohozaev-type identity, one shows the nonexistence of positive solutions of (1) when and with

The Proof Theorem 10 uses an identity of the Pohozaev type which we state in the following Lemma.

Lemma 9. Let be a positive solution of (1).
Then, the following identity is checked where

Proof. Multiplying Equation (1) by and integrating on , we obtain Set Calculation of According to the divergence theorem, However, from where one deduces from (36) and (37) that Calculation of
In [19], Mitidieri established the following relation: Let us apply to our case with Set Owing to the fact that on
we have Calculations of
We have by applying the divergence theorem, we obtain Calculations of
Applying Green’s formula generalized, we obtain therefore, From (43), (44), (46), and (48), we obtain Thus, From (39) and (50), we deduces that

Proof of Theorem 10 (By the absurdity). thus, we obtain Knowing that is strictly star-shaped then, what contradicts the fact that when .

5. Conclusion

In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem. Under some sufficient conditions on coefficients of equation of (1) such that , with is the critical Sobolev exponent, , and . In Section 3, we have shown the existence of at least one nontrivial solution in . In the Section 4, the nonexistence of positive solution at (1) has proved when can be a bounded, smooth, and stricly star-shaped domain, and with .

Data Availability

The author declares that the data supporting the findings of this study are available within the article.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

The author gratefully acknowledges the Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1397) during the academic year 1444AH/2023 AD.