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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:
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Figueiredo, On a class of nonlocal elliptic problmes with critical growth,, \emph{Differential equations and applications}, 2 (2010), 409. 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,, \emph{Nonlinear Analysis, 60 (2005), 611. 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,, \emph {Journal of Functional Analysis}, 122 (1994), 519. 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,, \emph{Communications in Contemporary Mathematics}, 16 (2014), 1450002. 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,, \emph{Journal of Differential Equations}, 193 (2003), 481. 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,, \emph{Journal of Differential Equations}, 250 (2011), 1876. 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,, \emph{Nonlinear Analysis, 75 (2012), 4496. 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,, \emph{Bulletin of the Australian Mathematical Society}, 77 (2006), 263. doi: 10.1017/S000497270003570X. Google Scholar [13] F. J. S. A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems,, \emph{Nonlinear Analysis, 59 (2004), 1147. 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,, \emph{Proceedings of Royal Society of Edinburgh Section A}, 127 (1997), 703. 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,, \emph{Calculus of Variations and Partial Differential Equations}, 3 (1995), 139. doi: 10.1007/BF01205003. Google Scholar [16] G. M. Figueiredo, Ground state soluttion for a Kirchhoff problem with exponential critical growth,, \emph{Milan Journal of Mathematics}, 84 (2016), 23. Google Scholar [17] F. Gazzola, Critical growth problems for polyharmonic operators,, \emph{Procedings of royal Society of edinberg Section A}, 128A (1998), 251. doi: 10.1017/S0308210500012774. Google Scholar [18] Y. Ge, J. Wei and F. Zhou, A critical elliptic problem for polyharmonic operator,, \emph{Journal of Functional Analysis}, 260 (2011), 2247. 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,, \emph{Revista de la Real Academia de Ciencias Exactas, 110 (2016), 219. Google Scholar [20] Sarika Goyal and K. Sreenadh, Existence of nontrivial solutions to quasilinear polyharmonic Kirchhoff equations with critical exponential growth,, \emph{Advances in Pure and Applied Mathematics}, 6 (2015), 1. 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,, \emph{Advances in Nonlinear Analysis}, 4 (2015), 177. doi: 10.1515/anona-2014-0034. Google Scholar [22] H. C. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, \emph{Calculas of Variations}, 3 (1995), 243. doi: 10.1007/BF01205006. Google Scholar [23] O. Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth,, \emph{Advances in Differential Equations}, 4 (1999), 877. Google Scholar [24] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equtions with subcritical and critical exponential growth,, \emph{Discrete and Continous Dynamical Systems}, 32 (2012), 2187. 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 $R^n$,, \emph{Journal of functional Analysis}, 262 (2012), 1132. 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,, \emph{Methods and Applications of Analysis}, 19 (2012), 243. 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,, \emph{Revista Matematica Iberoamericana}, 1 (1985), 185. 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 $R^n$,, \emph{Journal of Differential Equations}, 246 (2009), 1363. 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,, \emph{Differential Integral Equations}, 9 (1996), 967. Google Scholar [30] J. Moser, A sharp form of an inequality by N. Trudinger,, \emph{Indiana University Mathematics Journal}, 20 (1971), 1077. Google Scholar [31] R. Panda, Solution of a semilinear elliptic equation with critical growth in $\mathbb R^2$,, \emph{Nonlinear Analysis, 28 (1997), 721. 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 $R^2$,, \emph{Differential and Integral Equations}, 18 (2005), 681. Google Scholar [33] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, \emph{Journal de Matheatiques Pures et Appliqus}, 69 (1990), 55. Google Scholar [34] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, \emph{Annales de l'Institut Henri Poincare Analyse Non Linaire}, 9 (1992), 281. Google Scholar [35] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, \emph{Journal of Mathematical Analysis and Applications}, 318 (2006), 253. 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,, \emph{Journal of Functional Analysis}, 258 (2010), 99. 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,, \emph{Acta Mathematica Scientia}, 20 (2000), 547. Google Scholar

show all references

##### References:
 [1] D. R. Adams, A Sharp inequality of J. Moser for higher order derivatives,, \emph{Annals of Mathematics}, 128 (1988), 385. doi: 10.2307/1971445. Google Scholar [2] Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the $n$-Laplacian,, \emph{Annali della Scuola Normale Superiore di Pisa. Classe di Scienze}, 17 (1990), 393. Google Scholar [3] Adimurthi and K. Sandeep, A singular Moser-Trudinger embedding and its applications,, \emph{NoDEA Nonlinear Differential Equations and Applications}, 13 (2007), 585. 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,, \emph{International Mathematics Research Notices. IMRN}, 13 (2010), 2394. Google Scholar [5] C. O. Alves, F. Correa and G. M. Figueiredo, On a class of nonlocal elliptic problmes with critical growth,, \emph{Differential equations and applications}, 2 (2010), 409. 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,, \emph{Nonlinear Analysis, 60 (2005), 611. 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,, \emph {Journal of Functional Analysis}, 122 (1994), 519. 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,, \emph{Communications in Contemporary Mathematics}, 16 (2014), 1450002. 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,, \emph{Journal of Differential Equations}, 193 (2003), 481. 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,, \emph{Journal of Differential Equations}, 250 (2011), 1876. 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,, \emph{Nonlinear Analysis, 75 (2012), 4496. 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,, \emph{Bulletin of the Australian Mathematical Society}, 77 (2006), 263. doi: 10.1017/S000497270003570X. Google Scholar [13] F. J. S. A. Corrêa, On positive solutions of nonlocal and nonvariational elliptic problems,, \emph{Nonlinear Analysis, 59 (2004), 1147. 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,, \emph{Proceedings of Royal Society of Edinburgh Section A}, 127 (1997), 703. 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,, \emph{Calculus of Variations and Partial Differential Equations}, 3 (1995), 139. doi: 10.1007/BF01205003. Google Scholar [16] G. M. Figueiredo, Ground state soluttion for a Kirchhoff problem with exponential critical growth,, \emph{Milan Journal of Mathematics}, 84 (2016), 23. Google Scholar [17] F. Gazzola, Critical growth problems for polyharmonic operators,, \emph{Procedings of royal Society of edinberg Section A}, 128A (1998), 251. doi: 10.1017/S0308210500012774. Google Scholar [18] Y. Ge, J. Wei and F. Zhou, A critical elliptic problem for polyharmonic operator,, \emph{Journal of Functional Analysis}, 260 (2011), 2247. 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,, \emph{Revista de la Real Academia de Ciencias Exactas, 110 (2016), 219. Google Scholar [20] Sarika Goyal and K. Sreenadh, Existence of nontrivial solutions to quasilinear polyharmonic Kirchhoff equations with critical exponential growth,, \emph{Advances in Pure and Applied Mathematics}, 6 (2015), 1. 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,, \emph{Advances in Nonlinear Analysis}, 4 (2015), 177. doi: 10.1515/anona-2014-0034. Google Scholar [22] H. C. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents,, \emph{Calculas of Variations}, 3 (1995), 243. doi: 10.1007/BF01205006. Google Scholar [23] O. Lakkis, Existence of solutions for a class of semilinear polyharmonic equations with critical exponential growth,, \emph{Advances in Differential Equations}, 4 (1999), 877. Google Scholar [24] N. Lam and G. Lu, Existence of nontrivial solutions to polyharmonic equtions with subcritical and critical exponential growth,, \emph{Discrete and Continous Dynamical Systems}, 32 (2012), 2187. 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 $R^n$,, \emph{Journal of functional Analysis}, 262 (2012), 1132. 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,, \emph{Methods and Applications of Analysis}, 19 (2012), 243. 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,, \emph{Revista Matematica Iberoamericana}, 1 (1985), 185. 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 $R^n$,, \emph{Journal of Differential Equations}, 246 (2009), 1363. 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,, \emph{Differential Integral Equations}, 9 (1996), 967. Google Scholar [30] J. Moser, A sharp form of an inequality by N. Trudinger,, \emph{Indiana University Mathematics Journal}, 20 (1971), 1077. Google Scholar [31] R. Panda, Solution of a semilinear elliptic equation with critical growth in $\mathbb R^2$,, \emph{Nonlinear Analysis, 28 (1997), 721. 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 $R^2$,, \emph{Differential and Integral Equations}, 18 (2005), 681. Google Scholar [33] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators,, \emph{Journal de Matheatiques Pures et Appliqus}, 69 (1990), 55. Google Scholar [34] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, \emph{Annales de l'Institut Henri Poincare Analyse Non Linaire}, 9 (1992), 281. Google Scholar [35] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, \emph{Journal of Mathematical Analysis and Applications}, 318 (2006), 253. 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,, \emph{Journal of Functional Analysis}, 258 (2010), 99. 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,, \emph{Acta Mathematica Scientia}, 20 (2000), 547. Google Scholar
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