Multiple Solutions for a <i>p</i> (<i>x</i>)-Laplacian Problem: A Variational Approach

Khosravi Rashti, Mahnaz; Alimohammady, Mohsen; Ghobadi, Sirous

doi:https://doi.org/10.1155/2023/4261555

Journal of Mathematics

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Research Article | Open Access

Volume 2023 | Article ID 4261555 | https://doi.org/10.1155/2023/4261555

Multiple Solutions for a p (x)-Laplacian Problem: A Variational Approach

Mahnaz Khosravi Rashti,¹Mohsen Alimohammady,^1,2and Sirous Ghobadi¹

Academic Editor: Xiaolong Qin

Received17 Jun 2022

Revised09 Oct 2022

Accepted22 Feb 2023

Published23 Mar 2023

Abstract

We focus on an anisotropic p (x)-Laplacian equation defined on a bounded domain with smooth boundary. Moreover, using variational methods to find the existence of at least three solutions due to Ricceri is discussed.

1. Introduction

Here, the following is considered a Neumann boundary value problem:where and are non-negative and is its smooth boundary. We assume that is the outward normal vector on and are three Carathéodory functions. is a variable exponent and is regular enough. There are many articles, for example, [1–4] published in existence and multiplicity of solutions for different problems with different types of conditions.

Recently, the study of fourth-order boundary value problems involving Laplacian equations and p (x)-growth conditions was taken into consideration for many physical motivations and they can be modeled similar to this kind of equations (for example, see [5–11]). We also referred the reader to [12–23], wherein the existence of one, two, three, and infinity solutions for -Laplacian problems was studied by using variant variational methods and under some hypothesis on two nonlinear functions and for the parameters λ and which are in some intervals. For example, Fan and Deng in [8] studied the multiplicities of positive solutions for inhomogeneous Neumann problem of the kind -Laplacian as follows:

They considered the multiplicity of solutions for two cases in two cases .

Deng and Wang in [24] proved the existence and nonexistence of the following problem:

Heidarkhani and Salari in [25] by using the critical point theory discussed the existence of two and three solutions for the following problem:

Finally, in [26], Papageorgiou and Winkert proved the multiplicity of solutions for the following equation:

In this article, we would check the existence of at least three weak solutions for problem (1) when have a subcritical growth (conditions and ). Precisely, employing a three-critical point theorem due to Ricceri [27], we established the existing result for equations of -Laplacian type. Moreover, an example for illustrating our main result is given.

2. Basic Definitions and Preliminary Results

Given the Banach space , we denoted by the set of all maps which has the following properties.

Weakly converging of a sequence to in in which implies that has a subsequence strong convergence to .

Theorem 1. Assuming that is a separable reflexive real Banach space, a coercive map in which is -map and sequentially weakly lower semicontinuous, bounded on each bounded subset of , and has a continuous inverse on (dual of ). Suppose that is a -map such that is a compact function, has a strict local minimum such that .
SetIn the case that , for any compact interval (with convention ), there is such that for any and any map which is compact, there exists in which for any , the equation admits at least three weak solutions in in (ball with center 0 and radius ).
Consider , thenForwe set for , and we consider some conditions as follows: for almost all and all . There is a Carathéodory function which is continuously differentiable such that and for almost all . for almost all . There is such that the growth condition for almost all and all satisfy, where is the usual Euclidean norm. The monotonicity relation holds for almost all and each . This is equality if and only if . The inequalities hold for almost all and all . For in which for all , then Given such that for all and , then

The variable exponent Lebesgue space class is defined as follows:where is reflexive and separable Banach space endowed with the following Luxemburg norm (see [28]):

Moreover,is endowed with

We denoted as the closure of in .

Proposition 1 (see [28–31]). (1) is a reflexive and separable Banach space.(2) and for all , and there is a continuous embedding which is compact.(3) and for any , and there is a continuous embedding which is compact.(4)(Poincaré) , a number such that

Proposition 2. There exists a positive constant such that for all ,

Proof. For any ,Using the Hölder inequality, there is such thatHence,where and . Moreover,on is an equivalent to the one in equation (14), so for a suitable positive constant ,Set

Proposition 3 (see [32]). Given , the following properties hold:(1)(2)(3).Throughout this article, we assumed that and are two -Carathéodory function, i.e.,(a).(b).(c)and similarly for the function but with instead of .Corresponding to the functions and , we considered the maps and which are defined as follows:

Definition 1. A weak solution for equation (1) means thatwhere is the dimensional Hausdorff measure on the boundary.

3. Main Results

For and with , we setand , where

Define the functionals byfor each is well defined (see Theorem 3.1 in [23] and Proposition 4 in [15]), and it has continuous Gâteaux differentiable and it is sequentially weakly lower semicontinuous whose Gâteaux derivative is as follows:which admits a continuous inverse on the dual space . From and , and are continuously Gâteaux differentiable functions in which the embeddings and (Proposition 1) are compact which implies the compactness of the following derivatives:(for more details, see the proof of Theorem 1.1 in [23]).

Lemma 1. The functional is coercive and bounded on each bounded subset of .

Proof. For checking the coercivity of it is enough to assume that . From and Proposition 3,So, 𝒩 is coercive. Assume that is bounded, there is such that .
According to ,for some suitable positive constants and . Hence, on each bounded subset of , is bounded andMoreover, admits a strict local minimum with .

Lemma 2 (see Theorem 4.1 in [33]). Suppose that the Carathéodory function satisfies in and is of the type that is in and implies strongly in .

Lemma 3. lies in .

Proof. Let weakly in and . Since is weakly lower semicontinuous, . From the compact embedding (see Lemma 2.5 in [34]), converges to . Now, from Lemma 2, we need only to show thatSince , for a subsequence of which for simplicity we denoted it again by ,From and Lemmas 2.1 and 2.5 in [34],Thus, . According to Lemma 2, converges to strongly in , so .
Applying Lemmas 1and 3, we like to show the multiplicity of solutions.

Theorem 2. Suppose that There exists in which There exists a function such that

Then, for an interval , there exists in which “for every , there exists such that for each , problem (1) has at least three solutions in which whose norms in are less than .”

Proof. From , there exist and with such thatfor each is continuous on , and it is bounded on and , so choosing and leads to for all . Hence, from Proposition 2,for all . Hence, from equations (33) and (40),From equations (33) and (39) and Proposition 2, it follows that for each ,According to equations (41) and (42), the following relation is established:Assumption in conjunction with equations (43) implies the following equation:Thus, taking into account Lemmas 1 and 3, assumptions of Theorem 1 satisfy for and . Using Theorem 1, for interval , there exists such that for any , there exists and such that for each , problem (1) has at least three weak solutions whose norms in are less than (standard arguments show that is a Gâteaux differentiable functional and is a solution for problem (1) if and only if is a critical point of the function ).

Theorem 3. Letand

Then, for each interval , there exists in which for every , there exists such that for each , problem (1) has at least three solutions whose norms are less than in .

Proof. From equation (45), there exists an arbitrary and and with in which for any and any with . Since is continuous on , it is bounded on and . We can choose and in a similar manner that for all . By the same argument as in proof of Theorem 2, we have equations (41) and (42). Since is arbitrary so from equations (41) and (42),Then, using notation of Theorem 1, , and equation (46) implies . In this case, and . Finally, using Theorem 1, the proof is completed.
Now, we pay to the following example.

Example 1. Let and for every . Thus, for every , andAlso, by setting one has the following equation:However,Hence, Theorem 3 is applicable for the following system:For any and each , we put the following equation:where is introduced in the condition , . The next theorem provides sufficient conditions for applying Theorem 2 without satisfying in .

Theorem 4. Assume that the condition in Theorem 2 satifies and there exist , and so small such that for almost all , with and . Then, for each interval , there exists in which for every , there exists such that for each , problem (1) has at least three weak solutions in whose norms are less than .

Proof. Let be from and fix . We like to examine the validity of all assumptions of Theorem 2. We set the following equation:Directly, , for all and so . On the other hand, since for every ,So, we obtain for every . Moreover, the existence of continuous embedding.
(see Theorem 2.8 of [28]) implies the following equation:for positive constants , where is the volume of (for more details, see Theorem 2.11 [34]). Thus, . On the other hand, from , , and , one has the following equations:andSo, from equations (57) and (58) and , we obtain the following equation:Therefore, in Theorem 2 satisfies. Hence, Theorem 2 follows.
We studied the following problem to get results that the function has separated variables.where is a non-negative and nonzero function and is a continuous function. Set for every , where for all .
The following existing results are consequences of Theorem 4.

Corollary 1. Suppose that There is in which There exist , , and so small such thatfor all and

Then, for an interval (where and are the same as and in which is substituted by , respectively), there exists in which for any , there exists such that, for each , problem (60) has at least three weak solutions in whose norms are less than .

Theorem 5. Assume that there exists , , and so small such thatMoreover,

Then, for the given interval , where is the same as in which is replaced by , there exists such that for every , there exists such that, for each , problem (60) has at least three weak solutions in whose norms are less than .

Proof. From equation (65), satisfies for any . Moreover, using equation (64) and small enough, we can assume . Then, the conclusion follows from Corollary 1.

Remark 1. From cited results, we realized that nowhere in theorems asymptotic conditions on the functions and are required and only the algebraic assumptions on are supposed to give the existence of solutions.
The following example shows that nonlinearity verifies the hypotheses of Theorem 5.

Example 2. Let for all andThus, and is a continuous and non-negative function and by which choosing , we have the following equation: On the other hand, since , one has . Moreover,Hence, by applying Theorem 5 for each compact interval , there exists with the following property: for every , there exists in which for any , the problemhas at least three weak solutions whose norms in are less than .

Remark 2. Similar to the one in Proposition 2.13 in [34], we can consider and establish positive principal eigenvalues by following the -Laplacian Neumann problem:where and for almost all , and we discuss on number of solutions of problem (1) using problem (70) (precisely, to find out this the subject matter, see Theorem 2.14 in [34] and Theorem 4.4 in [35]).

Remark 3. In comparison with [23] when the authors worked with two parameters and , we obtained a precise range for the parameter λ in all the results of existence and multiplicity. We also noticed that this article differs from [13, 15, 16, 18, 19, 22, 34] either in the operators, in the boundary conditions, or in the nonlinearities of and . As noted, in the case of variable exponents, the question of multiplicity is recent, so our results contribute to these recent contributions by explaining the interval to the parameters and .

Conclusion 1. In this article, by using a variational method and Ricceri variational principle, we studied the -Laplacian equation under the Neumann boundary condition. We proved the existence and multiplicity solutions for problem (1), where , for , , and are Carathéodory functions and .

Data Availability

No real data were used to support this study. The data used in this study are hypothetical.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

P. Candito, U. Guarnotta, and K. Perera, “Two solutions for a parametric singular p-laplacian problem,” Journal of Nonlinear and Variational Analysis, vol. 4, pp. 455–468, 2020.
View at: Google Scholar
M. Kamenskii, N. Voskovskaya, and M. Zvereva, “On periodic oscillations of some points of a string with a nonlinear boundary condition,” Applied Set-Valued Analysis and Optimization, vol. 2, no. 1, pp. 35–48, 2020.
View at: Google Scholar
B. Ouhamou, A. Avouiil, and M. Berrajaa, “Existence and multiplicity results for discrete anisotropic equations,” J.Nonlinear Funct.Anal, vol. 20215, 2021.
View at: Google Scholar
Z. Yücedağ, “Solutions of nonlinear problems involving p(x)-Laplacian operator,” Adv. Nonlinear Anal., vol. 4, pp. 285–293, 2015.
View at: Publisher Site | Google Scholar
M. Alizadeh, M. Alimohammady, C. Cattani, and C. Cesarano, “On the minimal solution of bi-Laplacian equation,” Annals of the University of Craiova - Mathematics and Computer Science Series, vol. 46, no. 1, pp. 178–188, 2019.
View at: Google Scholar
Y. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.
View at: Publisher Site | Google Scholar
I. E. Eskandarkolaei, S. Khademloo, and G. Afrouzi, “Existence of weak solutions for second-order boundary-value problems of Kirchhoff-type with variable exponents,” Boletim da Sociedade Paranaense de Matemática, vol. 40, pp. 1–12, 2022.
View at: Publisher Site | Google Scholar
X. L. Fan and S. G. Deng, “Multiplicity of positive solutions for a class of inhomogeneous Neumann problems involving the p(x)-Laplacian,” Nonlinear Differential Equations and Applications NoDEA, vol. 16, no. 2, pp. 255–271, 2009.
View at: Publisher Site | Google Scholar
G. Fragnelli, “Positive periodic solutions for a system of anisotropic parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 73, pp. 110–121, 2010.
View at: Google Scholar
M. Ružička, Electro-rheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math, Springer, Berlin, Germany, 2000.
V. V. Zhikov, “Averaging of functionals of the calculus of variations and elasticity theory,” Math. USSR Izv. , vol. 29, no. 1, pp. 33–66, 1987.
View at: Publisher Site | Google Scholar
G. Barletta, A. Chinnì, and D. O'Regan, “Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities,” Nonlinear Anal. RWA. , vol. 27, pp. 312–325, 2016.
View at: Google Scholar
G. Bonanno and A. Chinnì, “Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent,” Journal of Mathematical Analysis and Applications, vol. 418, no. 2, pp. 812–827, 2014.
View at: Publisher Site | Google Scholar
G. Bonanno, G. Molica Bisci, and V. Rădulescu, “Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 14, pp. 4785–4795, 2011.
View at: Publisher Site | Google Scholar
M.-M. Boureanu and D.-N. Udrea, “Existence and multiplicity results for elliptic problems with p(.) -growth conditions,” Nonlinear Analysis: Real World Applications, vol. 14, no. 4, pp. 1829–1844, 2013.
View at: Publisher Site | Google Scholar
Y. H. Ki and K. Park, “Multiple solutions for equations of p(x)-Laplace type with nonlinear Neumann boundary condition,” Bulletin of the Korean Mathematical Society, vol. 53, no. 6, pp. 1805–1821, 2016.
View at: Publisher Site | Google Scholar
S. D. Lee, K. Park, and Y.-H. Kim, “Existence and multipicity of solutions for equations involving nonhomogeneous operators of p(x)-Laplace type in ℝ^N,” Bound Value Problem, vol. 2014, 2014.
View at: Publisher Site | Google Scholar
Q. Liu, “Existence of three solutions for p(x)-Laplacian equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 7, pp. 2119–2127, 2008.
View at: Publisher Site | Google Scholar
M. Mihăilescu, “Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1419–1425, 2007.
View at: Publisher Site | Google Scholar
S. Ouaro, A. Ouedraogo, and S. Soma, “Multivalued problem with Robin boundary condition involving diffuse measure data and variable exponent,” Advances in Nonlinear Analysis, vol. 3, no. 4, pp. 209–235, 2014.
View at: Publisher Site | Google Scholar
V. D. Rădulescu, “Nonlinear elliptic equations with variable exponent: old and new,” Nonlinear Analysis: Theory, Methods & Applications, vol. 121, pp. 336–369, 2015.
View at: Publisher Site | Google Scholar
X. Shi and X. Ding, “Existence and multiplicity of solutions for a general p(x)-Laplacian Neumann problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3715–3720, 2009.
View at: Publisher Site | Google Scholar
A. A. Zerouali, B. Karim, O. Chakrone, and A. Anane, “Existence and multiplicity results for elliptic problems with Nonlinear Boundary Conditions and variable exponents,” Boletim da Sociedade Paranaense de Matemática, vol. 33, no. 2, pp. 123–133, 2014.
View at: Publisher Site | Google Scholar
S. G. Deng and Q. Wang, “None existence, existence and multiplicity of positive solutions to the p(x)-Laplacian nonlinear Neumann boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 7, pp. 2170–2183, 2010.
View at: Publisher Site | Google Scholar
S. Heidarkhani and A. Salari, “p(x)-Laplacian-Like problems with Neumann condition originated from a capillary phenomena,” J.Nonlinear Funct. Anal. , vol. 2018, no. 11, pp. 1–23, 2018.
View at: Google Scholar
N. Papageorgiou and P. Winkert, “A Multiplicity Theorem for Anisotropic Robin Equations,” 2022, https://arxiv.org/abs/2103.06517.
View at: Google Scholar
B. Ricceri, “A further three critical points theorem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 4151–4157, 2009.
View at: Publisher Site | Google Scholar
O. Kovácik and J. Rákosnik, “On spaces L^p(x) and W^k,p(x),” Czechoslovak Mathematical Journal, vol. 41, pp. 592–618, 1991.
View at: Google Scholar
X. L. Fan, H. Q. Wu, and F. Z. Wang, “Hartman-type results for p(t)-Laplacian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 52, no. 2, pp. 585–594, 2003.
View at: Publisher Site | Google Scholar
X. L. Fan and D. Zhao, “On the generalized orlicz-Sobolev space W^k,p(x)(Ω),” J. Gancu Educ. College, vol. 12, no. 1, pp. 1–6, 1998.
View at: Google Scholar
P. Harjulehto, P. Hästö, M. Koskenoja, and S. Varonen, “The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values,” Potential Analysis, vol. 25, no. 3, pp. 205–222, 2006.
View at: Publisher Site | Google Scholar
X. L. Fan and X. Y. Han, “Existence and multiplicity of solutions for p(x)-Laplacian equations in ℝ^N,” Nonlinear Anal, vol. 59, pp. 173–188, 2004.
View at: Google Scholar
V. K. Le, “On a sub–supersolution method for variational inequalities with Leray–Lions operators in variable exponent spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3305–3321, 2009.
View at: Publisher Site | Google Scholar
I. H. Kim, Y.-H. Kim, and K. Park, “Existence of three solutions for equations of p(x)-Laplace type operators with nonlinear Neumann boundary conditions,” Boundary Value Problems, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar
J. Lee and Y.-H. Kim, “Multiplicity results for nonlinear Neumann boundary value problems involving p-Laplace type operators,” Boundary Value Problems, vol. 2016, no. 1, 2016.
View at: Publisher Site | Google Scholar

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Copyright © 2023 Mahnaz Khosravi Rashti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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