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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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [16] G. M. Figueiredo, Ground state soluttion for a Kirchhoff problem with exponential critical growth, Milan Journal of Mathematics, 84 (2016), 23-39. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [30] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana University Mathematics Journal, 20 (1971), 1077-1092. [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. [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. [33] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, Journal de Matheatiques Pures et Appliqus, 69 (1990), 55-83. [34] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Annales de l'Institut Henri Poincare Analyse Non Linaire, 9 (1992), 281-304. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [16] G. M. Figueiredo, Ground state soluttion for a Kirchhoff problem with exponential critical growth, Milan Journal of Mathematics, 84 (2016), 23-39. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [30] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana University Mathematics Journal, 20 (1971), 1077-1092. [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. [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. [33] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, Journal de Matheatiques Pures et Appliqus, 69 (1990), 55-83. [34] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Annales de l'Institut Henri Poincare Analyse Non Linaire, 9 (1992), 281-304. [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. [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. [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.
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