# American Institute of Mathematical Sciences

• Previous Article
A new proof of gradient estimates for mean curvature equations with oblique boundary conditions
• CPAA Home
• This Issue
• Next Article
Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$
September  2016, 15(5): 1689-1717. doi: 10.3934/cpaa.2016009

## Polyharmonic Kirchhoff type equations with singular exponential nonlinearities

 1 Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India, India, India

Received  September 2015 Revised  April 2016 Published  July 2016

In this article, we study the existence of non-negative solutions of the following polyharmonic Kirchhoff type problem with critical singular exponential nolinearity \begin{eqnarray} -M\left(\int_\Omega |\nabla^m u|^{\frac{n}{m}}dx\right)\Delta_{\frac{n}{m}}^{m} u = \frac{f(x,u)}{|x|^\alpha} \; \text{in}\; \Omega{,} \\ \quad u = \nabla u=\cdots= {\nabla}^{m-1} u=0 \quad \text{on} \quad \partial \Omega{,} \end{eqnarray} where $\Omega\subset R^n$ is a bounded domain with smooth boundary, $0 < \alpha < n$, $n\geq 2m\geq 2$ and $f(x,u)$ behaves like $e^{|u|^{\frac{n}{n-m}}}$ as $|u|\to\infty$. Using mountain pass structure and {the} concentration compactness principle, we show the existence of a nontrivial solution.
In the later part of the paper, we also discuss the above problem with convex-concave type sign changing nonlinearity. Using {the} Nehari manifold technique, we show the existence and multiplicity of non-negative solutions.
Citation: Pawan Kumar Mishra, Sarika Goyal, K. Sreenadh. Polyharmonic Kirchhoff type equations with singular exponential nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1689-1717. doi: 10.3934/cpaa.2016009
##### References:
 [1] D. R. Adams, A Sharp inequality of J. Moser for higher order derivatives, Annals of Mathematics, 128 (1988), 385-398. doi: 10.2307/1971445.  Google Scholar [2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 17 (1990), 393-413.  Google Scholar [3] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations and Applications, 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9.  Google Scholar [4] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in $\mathbb R^N$ and its applications, International Mathematics Research Notices. IMRN, 13 (2010), 2394-2426.  Google Scholar [5] C. O. Alves, F. Correa and G. M. Figueiredo, On a class of nonlocal elliptic problmes with critical growth, Differential equations and applications, 2 (2010), 409-417. doi: 10.7153/dea-02-25.  Google Scholar [6] C. O. Alves and A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problem, Nonlinear Analysis, Theory Methods and Applications, 60 (2005), 611-624. doi: 10.1016/j.na.2004.09.039.  Google Scholar [7] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, Journal of Functional Analysis, 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar [8] G. Autuori, F. Colasuonno and Patrizia Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Communications in Contemporary Mathematics, 16 (2014), 1450002-1450044. doi: 10.1142/S0219199714500023.  Google Scholar [9] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function, Journal of Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar [10] C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, Journal of Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar [11] F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Analysis, Theory Methods and Applications, 75 (2012), 4496-4512. doi: 10.1016/j.na.2011.09.048.  Google Scholar [12] F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of $p$-Kirchhoff-type via variational methods, Bulletin of the Australian Mathematical Society, 77 (2006), 263-277. doi: 10.1017/S000497270003570X.  Google Scholar [13] F. J. S. A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Analysis, Theory Methods and Applications, 59 (2004), 1147-1155. doi: 10.1016/j.na.2004.08.010.  Google Scholar [14] P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proceedings of Royal Society of Edinburgh Section A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar [15] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calculus of Variations and Partial Differential Equations, 3 (1995), 139-153. doi: 10.1007/BF01205003.  Google Scholar [16] G. M. Figueiredo, Ground state soluttion for a Kirchhoff problem with exponential critical growth, Milan Journal of Mathematics, 84 (2016), 23-39. Google Scholar [17] F. Gazzola, Critical growth problems for polyharmonic operators, Procedings of royal Society of edinberg Section A, 128A (1998), 251-263. doi: 10.1017/S0308210500012774.  Google Scholar [18] Y. Ge, J. Wei and F. Zhou, A critical elliptic problem for polyharmonic operator, Journal of Functional Analysis, 260 (2011), 2247-2282. doi: 10.1016/j.jfa.2011.01.005.  Google Scholar [19] Sarika Goyal, Pawan Mishra and K. Sreenadh, $n$-Kirchhoff type equations with exponential nonlinearities, Revista de la Real Academia de Ciencias Exactas, Ficas y Naturales. Serie A. Mathem icas, 110 (2016), 219-247. Google Scholar [20] Sarika Goyal and K. Sreenadh, Existence of nontrivial solutions to quasilinear polyharmonic Kirchhoff equations with critical exponential growth, Advances in Pure and Applied Mathematics, 6 (2015), 1-11. doi: 10.1515/apam-2014-0019.  Google Scholar [21] Sarika Goyal and K. Sreenadh, The Nehari manifold for a quasilinear polyharmonic equation with exponential nonlinearities and a sign-changing weight function, Advances in Nonlinear Analysis, 4 (2015), 177-200. doi: 10.1515/anona-2014-0034.  Google Scholar [22] H. C. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calculas of Variations, 3 (1995), 243-252. doi: 10.1007/BF01205006.  Google Scholar [23] O. Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth, Advances in Differential Equations, 4 (1999), 877-906.  Google Scholar [24] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equtions with subcritical and critical exponential growth, Discrete and Continous Dynamical Systems, 32 (2012), 2187-2205. doi: 10.3934/dcds.2012.32.2187.  Google Scholar [25] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $n$-Laplacian type with critical exponential growth in $R^n$, Journal of functional Analysis, 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012.  Google Scholar [26] N. Lam and G. Lu, Sharp singular Adams inequality in higher order sobolev spaces, Methods and Applications of Analysis, 19 (2012), 243-266. doi: 10.4310/MAA.2012.v19.n3.a2.  Google Scholar [27] P. L. Lions, The concentration compactness principle in the calculus of variations part-I, Revista Matematica Iberoamericana, 1 (1985), 185-201. doi: 10.4171/RMI/6.  Google Scholar [28] J. Marcos do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $R^n$, Journal of Differential Equations, 246 (2009), 1363-1386. doi: 10.1016/j.jde.2008.11.020.  Google Scholar [29] J. Marcus do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in $\Omega$ with nonlinearities in critical growth range, Differential Integral Equations, 9 (1996), 967-979.  Google Scholar [30] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana University Mathematics Journal, 20 (1971), 1077-1092.  Google Scholar [31] R. Panda, Solution of a semilinear elliptic equation with critical growth in $\mathbb R^2$, Nonlinear Analysis, Theory Methods and Applications, 28 (1997), 721-728. doi: 10.1016/0362-546X(95)00175-U.  Google Scholar [32] S. Prashanth and K. Sreenadh, Multiplicity of solutions to a nonhomogeneous elliptic equation in $R^2$, Differential and Integral Equations, 18 (2005), 681-698.  Google Scholar [33] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, Journal de Matheatiques Pures et Appliqus, 69 (1990), 55-83.  Google Scholar [34] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Annales de l'Institut Henri Poincare Analyse Non Linaire, 9 (1992), 281-304.  Google Scholar [35] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, Journal of Mathematical Analysis and Applications, 318 (2006), 253-270. doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar [36] T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\Omega$ involving sign-changing weight, Journal of Functional Analysis, 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.  Google Scholar [37] X. Zheng and Y. Deng, Existence of multiple solutions for a semilinear biharmonic equation with critical exponent, Acta Mathematica Scientia, 20 (2000), 547-554.  Google Scholar

show all references

##### References:
 [1] D. R. Adams, A Sharp inequality of J. Moser for higher order derivatives, Annals of Mathematics, 128 (1988), 385-398. doi: 10.2307/1971445.  Google Scholar [2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze, 17 (1990), 393-413.  Google Scholar [3] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications, NoDEA Nonlinear Differential Equations and Applications, 13 (2007), 585-603. doi: 10.1007/s00030-006-4025-9.  Google Scholar [4] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger-Moser inequality in $\mathbb R^N$ and its applications, International Mathematics Research Notices. IMRN, 13 (2010), 2394-2426.  Google Scholar [5] C. O. Alves, F. Correa and G. M. Figueiredo, On a class of nonlocal elliptic problmes with critical growth, Differential equations and applications, 2 (2010), 409-417. doi: 10.7153/dea-02-25.  Google Scholar [6] C. O. Alves and A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problem, Nonlinear Analysis, Theory Methods and Applications, 60 (2005), 611-624. doi: 10.1016/j.na.2004.09.039.  Google Scholar [7] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, Journal of Functional Analysis, 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.  Google Scholar [8] G. Autuori, F. Colasuonno and Patrizia Pucci, On the existence of stationary solutions for higher-order p-Kirchhoff problems, Communications in Contemporary Mathematics, 16 (2014), 1450002-1450044. doi: 10.1142/S0219199714500023.  Google Scholar [9] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign-changing weight function, Journal of Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar [10] C. Chen, Y. Kuo and T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, Journal of Differential Equations, 250 (2011), 1876-1908. doi: 10.1016/j.jde.2010.11.017.  Google Scholar [11] F. Colasuonno, P. Pucci and C. Varga, Multiple solutions for an eigenvalue problem involving p-Laplacian type operators, Nonlinear Analysis, Theory Methods and Applications, 75 (2012), 4496-4512. doi: 10.1016/j.na.2011.09.048.  Google Scholar [12] F. J. S. A. Corrêa and G. M. Figueiredo, On an elliptic equation of $p$-Kirchhoff-type via variational methods, Bulletin of the Australian Mathematical Society, 77 (2006), 263-277. doi: 10.1017/S000497270003570X.  Google Scholar [13] F. J. S. A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Analysis, Theory Methods and Applications, 59 (2004), 1147-1155. doi: 10.1016/j.na.2004.08.010.  Google Scholar [14] P. Drabek and S. I. Pohozaev, Positive solutions for the p-Laplacian: application of the fibering method, Proceedings of Royal Society of Edinburgh Section A, 127 (1997), 703-726. doi: 10.1017/S0308210500023787.  Google Scholar [15] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calculus of Variations and Partial Differential Equations, 3 (1995), 139-153. doi: 10.1007/BF01205003.  Google Scholar [16] G. M. Figueiredo, Ground state soluttion for a Kirchhoff problem with exponential critical growth, Milan Journal of Mathematics, 84 (2016), 23-39. Google Scholar [17] F. Gazzola, Critical growth problems for polyharmonic operators, Procedings of royal Society of edinberg Section A, 128A (1998), 251-263. doi: 10.1017/S0308210500012774.  Google Scholar [18] Y. Ge, J. Wei and F. Zhou, A critical elliptic problem for polyharmonic operator, Journal of Functional Analysis, 260 (2011), 2247-2282. doi: 10.1016/j.jfa.2011.01.005.  Google Scholar [19] Sarika Goyal, Pawan Mishra and K. Sreenadh, $n$-Kirchhoff type equations with exponential nonlinearities, Revista de la Real Academia de Ciencias Exactas, Ficas y Naturales. Serie A. Mathem icas, 110 (2016), 219-247. Google Scholar [20] Sarika Goyal and K. Sreenadh, Existence of nontrivial solutions to quasilinear polyharmonic Kirchhoff equations with critical exponential growth, Advances in Pure and Applied Mathematics, 6 (2015), 1-11. doi: 10.1515/apam-2014-0019.  Google Scholar [21] Sarika Goyal and K. Sreenadh, The Nehari manifold for a quasilinear polyharmonic equation with exponential nonlinearities and a sign-changing weight function, Advances in Nonlinear Analysis, 4 (2015), 177-200. doi: 10.1515/anona-2014-0034.  Google Scholar [22] H. C. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calculas of Variations, 3 (1995), 243-252. doi: 10.1007/BF01205006.  Google Scholar [23] O. Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth, Advances in Differential Equations, 4 (1999), 877-906.  Google Scholar [24] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equtions with subcritical and critical exponential growth, Discrete and Continous Dynamical Systems, 32 (2012), 2187-2205. doi: 10.3934/dcds.2012.32.2187.  Google Scholar [25] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of $n$-Laplacian type with critical exponential growth in $R^n$, Journal of functional Analysis, 262 (2012), 1132-1165. doi: 10.1016/j.jfa.2011.10.012.  Google Scholar [26] N. Lam and G. Lu, Sharp singular Adams inequality in higher order sobolev spaces, Methods and Applications of Analysis, 19 (2012), 243-266. doi: 10.4310/MAA.2012.v19.n3.a2.  Google Scholar [27] P. L. Lions, The concentration compactness principle in the calculus of variations part-I, Revista Matematica Iberoamericana, 1 (1985), 185-201. doi: 10.4171/RMI/6.  Google Scholar [28] J. Marcos do Ó, E. Medeiros and U. Severo, On a quasilinear nonhomogeneous elliptic equation with critical growth in $R^n$, Journal of Differential Equations, 246 (2009), 1363-1386. doi: 10.1016/j.jde.2008.11.020.  Google Scholar [29] J. Marcus do Ó, Semilinear Dirichlet problems for the $N$-Laplacian in $\Omega$ with nonlinearities in critical growth range, Differential Integral Equations, 9 (1996), 967-979.  Google Scholar [30] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana University Mathematics Journal, 20 (1971), 1077-1092.  Google Scholar [31] R. Panda, Solution of a semilinear elliptic equation with critical growth in $\mathbb R^2$, Nonlinear Analysis, Theory Methods and Applications, 28 (1997), 721-728. doi: 10.1016/0362-546X(95)00175-U.  Google Scholar [32] S. Prashanth and K. Sreenadh, Multiplicity of solutions to a nonhomogeneous elliptic equation in $R^2$, Differential and Integral Equations, 18 (2005), 681-698.  Google Scholar [33] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, Journal de Matheatiques Pures et Appliqus, 69 (1990), 55-83.  Google Scholar [34] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Annales de l'Institut Henri Poincare Analyse Non Linaire, 9 (1992), 281-304.  Google Scholar [35] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, Journal of Mathematical Analysis and Applications, 318 (2006), 253-270. doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar [36] T. F. Wu, Multiple positive solutions for a class of concave-convex elliptic problems in $\Omega$ involving sign-changing weight, Journal of Functional Analysis, 258 (2010), 99-131. doi: 10.1016/j.jfa.2009.08.005.  Google Scholar [37] X. Zheng and Y. Deng, Existence of multiple solutions for a semilinear biharmonic equation with critical exponent, Acta Mathematica Scientia, 20 (2000), 547-554.  Google Scholar
 [1] Nguyen Lam. Equivalence of sharp Trudinger-Moser-Adams Inequalities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 973-998. doi: 10.3934/cpaa.2017047 [2] Kyril Tintarev. Is the Trudinger-Moser nonlinearity a true critical nonlinearity?. Conference Publications, 2011, 2011 (Special) : 1378-1384. doi: 10.3934/proc.2011.2011.1378 [3] Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006 [4] Djairo G. De Figueiredo, João Marcos do Ó, Bernhard Ruf. Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 455-476. doi: 10.3934/dcds.2011.30.455 [5] Shiqiu Fu, Kanishka Perera. On a class of semipositone problems with singular Trudinger-Moser nonlinearities. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1747-1756. doi: 10.3934/dcdss.2020452 [6] Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121 [7] Federica Sani. A biharmonic equation in $\mathbb{R}^4$ involving nonlinearities with critical exponential growth. Communications on Pure & Applied Analysis, 2013, 12 (1) : 405-428. doi: 10.3934/cpaa.2013.12.405 [8] Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683 [9] Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011 [10] Chao Ji, Vicenţiu D. Rădulescu. Concentration phenomena for magnetic Kirchhoff equations with critical growth. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021088 [11] Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2021038 [12] Erisa Hasani, Kanishka Perera. On the compactness threshold in the critical Kirchhoff equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021106 [13] Quanqing Li, Kaimin Teng, Xian Wu. Ground states for Kirchhoff-type equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2623-2638. doi: 10.3934/cpaa.2018124 [14] Yinbin Deng, Wentao Huang. Least energy solutions for fractional Kirchhoff type equations involving critical growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1929-1954. doi: 10.3934/dcdss.2019126 [15] Shuangjie Peng. Remarks on singular critical growth elliptic equations. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 707-719. doi: 10.3934/dcds.2006.14.707 [16] Rui-Qi Liu, Chun-Lei Tang, Jia-Feng Liao, Xing-Ping Wu. Positive solutions of Kirchhoff type problem with singular and critical nonlinearities in dimension four. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1841-1856. doi: 10.3934/cpaa.2016006 [17] Van Hoang Nguyen. The Hardy–Moser–Trudinger inequality via the transplantation of Green functions. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3559-3574. doi: 10.3934/cpaa.2020155 [18] Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110 [19] Prosenjit Roy. On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5207-5222. doi: 10.3934/dcds.2019212 [20] Kanishka Perera, Marco Squassina. Bifurcation results for problems with fractional Trudinger-Moser nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 561-576. doi: 10.3934/dcdss.2018031

2020 Impact Factor: 1.916