Abstract
In this study, we prove the existence of a positive solution for a -Kirchhoff-type problem with Sobolev exponent.
1. Introduction and Main Results
In this study, we are concerned with the following -Kirchhoff-type problem:where is a bounded domain, , , , , is the usual norm in given by , is a parameter, and is the critical Sobolev exponent corresponding to the noncompact embedding of into .
Since equation contains an integral over , it is no longer a pointwise identity; therefore, it is often called a nonlocal problem. It is called also nondegenerate if and , while it is named degenerate if and .
Such nonlocal elliptic problem such as is related to the original Kirchhoff’s equation in [1] which was first introduced by Kirchhoff as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the strings produced by transverse vibrations.
Much interest has grown on problems involving critical exponents, starting from the celebrated paper by Brézis and Nirenberg [2]. They considered problem with , , and . From their results, it came out that the space dimension was going to play a crucial role. They established existence results in dimension when is a ball, namely, they ensure the existence of a positive constant such that problem admits a positive solution for , where is the first eigenvalue of the operator . In higher dimensions, , they proved the existence of a positive solution for and no positive solution for or and is a starshaped domain.
Moreover, by using the concentration compactness principle [3], the results of [2] were extended to the quasi-linear cases by Guedda and Veron [4]. Precisely, they proved that if , then the quasi-linear Brezis–Nirenberg problem,has a positive solution if , where is the first eigenvalue ofwith Dirichlet boundary condition and no positive solution for or and star shaped.
In the last few years, great attention has been paid to the study of Kirchhoff problems involving critical nonlinearities. This problems create many difficulties in applying variational methods, we refer the readers to [5–13] and the references therein. More precisely, Naimen in [10] generalized the results of [2] for 3- and 4-dimensional Kirchhoff-type equations. For larger dimensional case, Figueiredo [7] considers the case if is sufficiently large.
The main results in the present paper can be considered as the extension of the work of [4] for a p-Kirchhoff problem with large range of . The competing effect of the nonlocal term with the critical nonlinearity and the lack of compactness of the embedding of into , prevents us from using the variational methods in a standard way. So, we need more delicate estimates.
To the best of our knowledge, many of the results are new for and even in the case . Our results and setting are more general and delicate, it is difficult to obtain the solution in the degenerate case when .
Our technique is based on variational methods and concentration compactness argument [3], and we need to estimate the energy levels.
In this paper, we define the best Sobolev constant for the imbedding ↪ as
Then, we obtain the following existence result.
Theorem 1. Let , , and . If and or and , then has a positive solution.
Remark 1. If is a solution of , we obtainwhere with and . By the extended Pohozaev identity in [4], we can get a nonexistence solutions for and is starshaped.
2. Preliminary Results
In this study, we use the following notation: (resp ) denotes strong (resp. weak) convergence, denotes as , is the first eigenvalue ofwith Dirichlet boundary condition, and is the ball centered at and of radius , .
Recall that the infimum is attained in by the functions of the form
Moreover, satisfies
Let be a positive constant and set such that for and for and . Set , by taking so that .
We have the well-known estimates as :(see [14]).
The energy functional , corresponding to the problem , is given by
Notice that is well defined in and belongs to . We say that is a weak solution of if, for any , there holds
Hence, a critical point of functional is a weak solution of problem .
Definition 1. Let ; a sequence is called a sequence (Palais–Smale sequence at level ) ifLet . We say that satisfies the Palais–Smale condition at level if any sequence contains a convergent subsequence in .
Lemma 1. Let , , , and , for . For , we consider the function , given by
Then,(1)If , , and , then the equation has a unique positive solution, and , for all .(2)If , then the equation has a unique positive solution and , for all .
Proof. (1)For , , and , we have that is, the equation has a unique positive solution, and , for all .(2)For , we have andThen, , , for , and , for . Hence, is concave function andMoreover, we have and ; thus, from (18) and the concavity of , we can conclude that the equation has a unique positive solution and , for all .
Now, we will verify that the functional exhibits the Mountain Pass geometry.
Lemma 2. Assume , , and . Suppose that and or and . Then, there exist positive numbers and such that, for all , we have(1), with (2)There exists such that and
Proof. (1)Let ; by Sobolev and Young inequalities, it holds that Let ; since , , and and from (19), one has Let us define Then,(i)Let and . We have is strictly increasing on the interval: and it is strictly decreasing on the interval with , , and Then, direct calculation shows that So, for all , we have(ii)When , one obtains . Let ; from (20), one has If , similar to (i), there exist , such that(2)For , , we have(i)If and , then . So, we can easily find with , such that .(ii)If and , using (9) and taking small enough, thenfor all . Then, it follows from the above inequality, as . Thus, choosing sufficiently large and putting , we have a function satisfying , such that .
Proof. The proof is complete.
Lemma 3. Assume , , , and . Let and be a sequence for ; then, there exists a subsequence of which we still denote by and such thatwith .
Proof. We havethat is,for any .
Then, as , it follows thatAs , we obtain that is bounded in . Up to a subsequence if necessary, there exists a function such thatThen,and thus, . This completes the proof of Lemma 4.
Now, we prove an important lemma which ensures the local compactness of the Palais–Smale sequence for .
For , is defined in Lemma 2, and we define
Lemma 4. Let and , and suppose and or and . If is a sequence for with , then contains a subsequence converging strongly in .
Proof. By the proof of Lemma 3, we have is a bounded sequence in . Hence, by the concentration compactness principle due to Lions [3], there exists a subsequence, still denoted by , such thatwhere is an at most countable index set, is the Dirac mass at and , and are sets of nonnegative real numbers. Moreover,For , let be a smooth cut-off function centered at such that , andSince is bounded in and as , it holds by Hölder’s inequality:Then, . Therefore, by (41), we deduce thatAssume by contradiction that there exists such that . Set and ; then, by (44), we obtainIt is clear that , thanks to . So, from (45) and the definition of in Lemma 1, we obtainWe will discuss it in two cases: Case 1: and . According to Lemma 1, we have and if with which implies that Case 2: and . In this case, from Lemma 2, we get and if withwhich implies thatHence, using (41), we deduceBy Young inequality, we haveWe observe that , , and thus, for , we obtainsince for and is defined in (39). It is a contradiction with . Then, is empty, which implies thatNow, set as ; then, we havefor any .
Let ; then, from (55) and (56), we deduce thatTaking the test function in (59), we obtainTherefore, equalities (58) and (59) imply that . Consequently converges strongly in , which is the desired result.
2.1. Proof of the Main Result
By Lemma 4, satisfies the Palais–Smale condition at level for any . So, the existence of the positive solution follows immediately from the following estimates.
Lemma 5. Let , , and . Suppose that and or and . Then,
Proof. Employing estimate (9), we define the following functions:Note that and when is close to 0 so that is attained for some . Furthermore, from , it follows thatand therefore,Choose small enough so that, by (9), we have , for some .
Besides, it holdsFor , , and , we have by (9)Then, for small enough, the above estimates yield for some (independently of ).
For , , and and for small enough, we have, by (63),which implies that is bounded above, for all , that is, there exists a positive real number (independently of ).
Now, we estimate . It follows from thatthat is,Set , , andThen, by (69) and the definition of , we obtainwhich implies from the proof of Lemma 1 that . Therefore, , where . As is concave, then is convex, soBy , we haveSo, from (73), we deduce thatConsequently, by (9),which is the desired result.
Now, we can proof the existence of a Mountain Pass-type solution.
Proof of Theorem 1. Note that , so from Lemma 2, satisfies the geometry conditions of the Mountain Pass Theorem [15]. Then, there exists a Palais–Smale sequence at level , such thatwithwhereUsing Lemma 3, we have that has a subsequence, still denoted by , such that in as . Hence, from Lemma 4 and 5, we have in as . Hence, and . So, as , we can conclude that is a nonzero solution of with positive energy. Now, we show that becausewhich implies that . By the strong maximum principle [16], one has . This completes the Proof of Theorem 1.
Data Availability
The functions, functionals, and parameters used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors gratefully acknowledge Qassim University, represented by the Deanship of Scientific Research, on the material support for this research under the number (1121) during the academic year 1443AH/2021AD and Algerian Ministry of Higher Education and Scientific Research on the material support for this research under the number (1121) during the academic year 1443AH/2021AD.