# American Institute of Mathematical Sciences

April  2012, 32(4): 1095-1124. doi: 10.3934/dcds.2012.32.1095

## A convex-concave elliptic problem with a parameter on the boundary condition

 1 Dpto. de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna 2 Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, 03080 Alicante, Spain 3 Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 – La Laguna, Spain

Received  September 2010 Revised  July 2011 Published  October 2011

In this paper we study existence and multiplicity of nonnegative solutions to $$$$\left\{\begin{array}{ll} \Delta u = u^p + u^q \qquad & \mbox{in }\Omega, \\ \frac{\partial u }{\partial \nu} =\lambda u \qquad & \mbox{on }\partial \Omega. \end{array}\right.$$$$ Here $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, $\nu$ stands for the outward unit normal and $p$, $q$ are in the convex-concave case, that is $0 < q < 1 < p$. We prove that there exists $\Lambda^* >0$ such that there are no nonnegative solutions for $\lambda < \Lambda^*$, and there is a maximal nonnegative solution for $\lambda \ge \Lambda^{*}$. If $\lambda$ is large enough, then there exist at least two nonnegative solutions. We also study the asymptotic behavior of solutions when $\lambda\to \infty$ and the occurrence of dead cores. In the particular case where $\Omega$ is the unit ball of $\mathbb{R}^N$ we show exact multiplicity of radial nonnegative solutions when $\lambda$ is large enough, and also the existence of nonradial nonnegative solutions.
Citation: J. García-Melián, Julio D. Rossi, José Sabina de Lis. A convex-concave elliptic problem with a parameter on the boundary condition. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1095-1124. doi: 10.3934/dcds.2012.32.1095
##### References:
 [1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519. doi: 10.1006/jfan.1994.1078. Google Scholar [2] A. Ambrosetti, J. García-Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations,, J. Funct. Anal., 137 (1996), 219. doi: 10.1006/jfan.1996.0045. Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [4] D. Arcoya and J. D. Rossi, Antimaximum principle for quasilinear problems,, Adv. Diff. Eqns., 9 (2004), 1185. Google Scholar [5] J. M. Arrieta, R. Pardo and A. Rodríguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 225. Google Scholar [6] J. M. Arrieta, R. Pardo and A. Rodríguez-Bernal, Equilibria and global dynamics of a problem with bifurcation from infinity,, J. Diff. Eqns., 246 (2009), 2055. doi: 10.1016/j.jde.2008.09.002. Google Scholar [7] C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour,, J. Anal. Math., 58 (1992), 9. doi: 10.1007/BF02790355. Google Scholar [8] F. Charro, E. Colorado and I. Peral, Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side,, J. Diff. Eqns., 246 (2009), 4221. doi: 10.1016/j.jde.2009.01.013. Google Scholar [9] E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions,, J. Funct. Anal., 199 (2003), 468. doi: 10.1016/S0022-1236(02)00101-5. Google Scholar [10] M. Del Pino and C. Flores, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains,, Comm. Partial Differential Equations, 26 (2001), 2189. Google Scholar [11] J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term,, Trans. Amer. Math. Soc., 323 (1991), 877. doi: 10.2307/2001562. Google Scholar [12] J. García Azorero and I. Peral Alonso, Some results about existence of a second positive solution in a quasilinear critical problem,, Indiana Univ. Math. J., 43 (1994), 941. doi: 10.1512/iumj.1994.43.43041. Google Scholar [13] J. García Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition,, J. Diff. Eqns., 198 (2004), 91. Google Scholar [14] J. García-Melián, Uniqueness of positive solutions for a boundary blow-up problem,, J. Math. Anal. Appl., 360 (2009), 530. doi: 10.1016/j.jmaa.2009.06.077. Google Scholar [15] J. García-Melián, J. D. Rossi and J. Sabina de Lis, A bifurcation problem governed by the boundary condition. I,, NoDEA Nonlinear Differential Equations and Applications, 14 (2007), 499. Google Scholar [16] J. García-Melián, J. D. Rossi and J. Sabina de Lis, A bifurcation problem governed by the boundary condition. II,, Proc. London Math. Soc. (3), 94 (2007), 1. Google Scholar [17] J. García-Melián, J. D. Rossi and J. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions,, Comm. Contemp. Math., 11 (2009), 585. doi: 10.1142/S0219199709003508. Google Scholar [18] J. García-Melián, J. D. Rossi and J. Sabina de Lis, An elliptic system with bifurcation parameters on the boundary condition,, J. Diff. Eqns., 247 (2009), 779. Google Scholar [19] J. García-Melián, J. D. Rossi and J. Sabina de Lis, Layer profiles of solutions to elliptic problems under parameter-dependent boundary conditions,, Zeitschrift für Analysis und ihre Anwendungen (Journal for Analysis and its Applications), 29 (2010), 451. Google Scholar [20] J. García-Melián and J. Sabina de Lis, Uniqueness to quasilinear problems for the p-Laplacian in radially symmetric domains,, Nonlinear Anal. Ser. A: Theory Methods, 43 (2001), 803. Google Scholar [21] J. García-Melián and J. Sabina de Lis, Remarks on large solutions,, in, (2005), 31. Google Scholar [22] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 321. Google Scholar [23] P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (1964). Google Scholar [24] J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402. Google Scholar [25] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Regional Conf. Ser. in Math., 65 (1986). Google Scholar [26] W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian,, J. Inequal. Appl., 1 (1997), 47. doi: 10.1155/S1025583497000040. Google Scholar [27] J. D. Rossi, Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem,, in, (2005), 311. Google Scholar

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##### References:
 [1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, J. Funct. Anal., 122 (1994), 519. doi: 10.1006/jfan.1994.1078. Google Scholar [2] A. Ambrosetti, J. García-Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations,, J. Funct. Anal., 137 (1996), 219. doi: 10.1006/jfan.1996.0045. Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications,, J. Funct. Anal., 14 (1973), 349. doi: 10.1016/0022-1236(73)90051-7. Google Scholar [4] D. Arcoya and J. D. Rossi, Antimaximum principle for quasilinear problems,, Adv. Diff. Eqns., 9 (2004), 1185. Google Scholar [5] J. M. Arrieta, R. Pardo and A. Rodríguez-Bernal, Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 225. Google Scholar [6] J. M. Arrieta, R. Pardo and A. Rodríguez-Bernal, Equilibria and global dynamics of a problem with bifurcation from infinity,, J. Diff. Eqns., 246 (2009), 2055. doi: 10.1016/j.jde.2008.09.002. Google Scholar [7] C. Bandle and M. Marcus, "Large" solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour,, J. Anal. Math., 58 (1992), 9. doi: 10.1007/BF02790355. Google Scholar [8] F. Charro, E. Colorado and I. Peral, Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side,, J. Diff. Eqns., 246 (2009), 4221. doi: 10.1016/j.jde.2009.01.013. Google Scholar [9] E. Colorado and I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions,, J. Funct. Anal., 199 (2003), 468. doi: 10.1016/S0022-1236(02)00101-5. Google Scholar [10] M. Del Pino and C. Flores, Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains,, Comm. Partial Differential Equations, 26 (2001), 2189. Google Scholar [11] J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term,, Trans. Amer. Math. Soc., 323 (1991), 877. doi: 10.2307/2001562. Google Scholar [12] J. García Azorero and I. Peral Alonso, Some results about existence of a second positive solution in a quasilinear critical problem,, Indiana Univ. Math. J., 43 (1994), 941. doi: 10.1512/iumj.1994.43.43041. Google Scholar [13] J. García Azorero, I. Peral and J. D. Rossi, A convex-concave problem with a nonlinear boundary condition,, J. Diff. Eqns., 198 (2004), 91. Google Scholar [14] J. García-Melián, Uniqueness of positive solutions for a boundary blow-up problem,, J. Math. Anal. Appl., 360 (2009), 530. doi: 10.1016/j.jmaa.2009.06.077. Google Scholar [15] J. García-Melián, J. D. Rossi and J. Sabina de Lis, A bifurcation problem governed by the boundary condition. I,, NoDEA Nonlinear Differential Equations and Applications, 14 (2007), 499. Google Scholar [16] J. García-Melián, J. D. Rossi and J. Sabina de Lis, A bifurcation problem governed by the boundary condition. II,, Proc. London Math. Soc. (3), 94 (2007), 1. Google Scholar [17] J. García-Melián, J. D. Rossi and J. Sabina de Lis, Existence and uniqueness of positive solutions to elliptic problems with sublinear mixed boundary conditions,, Comm. Contemp. Math., 11 (2009), 585. doi: 10.1142/S0219199709003508. Google Scholar [18] J. García-Melián, J. D. Rossi and J. Sabina de Lis, An elliptic system with bifurcation parameters on the boundary condition,, J. Diff. Eqns., 247 (2009), 779. Google Scholar [19] J. García-Melián, J. D. Rossi and J. Sabina de Lis, Layer profiles of solutions to elliptic problems under parameter-dependent boundary conditions,, Zeitschrift für Analysis und ihre Anwendungen (Journal for Analysis and its Applications), 29 (2010), 451. Google Scholar [20] J. García-Melián and J. Sabina de Lis, Uniqueness to quasilinear problems for the p-Laplacian in radially symmetric domains,, Nonlinear Anal. Ser. A: Theory Methods, 43 (2001), 803. Google Scholar [21] J. García-Melián and J. Sabina de Lis, Remarks on large solutions,, in, (2005), 31. Google Scholar [22] N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 321. Google Scholar [23] P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (1964). Google Scholar [24] J. B. Keller, On solutions of $\Delta u=f(u)$,, Comm. Pure Appl. Math., 10 (1957), 503. doi: 10.1002/cpa.3160100402. Google Scholar [25] P. H. Rabinowitz, "Minimax Methods in Critical Point Theory with Applications to Differential Equations,", CBMS Regional Conf. Ser. in Math., 65 (1986). Google Scholar [26] W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian,, J. Inequal. Appl., 1 (1997), 47. doi: 10.1155/S1025583497000040. Google Scholar [27] J. D. Rossi, Elliptic problems with nonlinear boundary conditions and the Sobolev trace theorem,, in, (2005), 311. Google Scholar
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