Concentrating solutions for an anisotropic elliptic problem with large exponent. (English) Zbl 1320.35081
This paper studies the boundary value problem
\[
\nabla (a(x)\nabla u)+a(x)u^p=0,\;\;\;u(x)>0\;\;x\in \Omega,\;\;\;u(x)=0\;\;x\in\partial\Omega,
\]
where \(\Omega\subset \mathbb R^2\) is a smooth bounded domain, and \(a(x)\) is a smooth positive function on \(\overline{\Omega}\). The focus here is on the limit \(p\rightarrow \infty\). Previous studies concern the case in which \(a(x)\) is constant. Here new phenomena are revealed in the case in which \(a(x)\) is nonconstant. The main result shows that, given a local maximum point of the function \(a(x)\), there exists an arbitrarily large number of solutions as \(p\rightarrow \infty\), concentrating near the maximum point. The proof employs a combination of Lyapunov-Schmidt reduction and variational methods.
Reviewer: Guy Katriel (Haifa)
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
References:
| [1] | Adimurthi, Asymptotic estimates for a two-dimensional problem with polynomial nonlinearity,, Proc. Amer. Math. Soc., 132, 1013 (2004) · Zbl 1083.35035 · doi:10.1090/S0002-9939-03-07301-5 |
| [2] | T. Aubin, <em>Nonlinear Analysis on Manifolds. Monge-Ampère equations</em>,, Springer-Verlag (1982) · Zbl 0512.53044 · doi:10.1007/978-1-4612-5734-9 |
| [3] | A. Bahri, On a variational problem with lack of compactness: The topological effect of the critical points at infinity,, Calc. Var. Partial Differential Equations, 3, 67 (1995) · Zbl 0814.35032 · doi:10.1007/BF01190892 |
| [4] | P. Bates, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability,, Adv. Diff. Equations, 4, 1 (1999) · Zbl 1157.35407 |
| [5] | D. Chae, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory,, Comm. Math. Phys., 215, 119 (2000) · Zbl 1002.58015 · doi:10.1007/s002200000302 |
| [6] | C. C. Chen, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces,, Comm. Pure Appl. Math. 55 (2002), 55, 728 (2002) · Zbl 1040.53046 · doi:10.1002/cpa.3014 |
| [7] | W. Chen, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63, 615 (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8 |
| [8] | E. N. Dancer, Multipeak solutions for a singular perturbed Neumann problem,, Pacific J. Math., 189, 241 (1999) · Zbl 0933.35070 · doi:10.2140/pjm.1999.189.241 |
| [9] | M. del Pino, Two-bubble solutions in the super-critical Bahri-Coron’s problem,, Calc. Var. Partial Differential Equations, 16, 113 (2003) · Zbl 1142.35421 · doi:10.1007/s005260100142 |
| [10] | M. del Pino, Singular limits in Liouville-type equations,, Calc. Var. Partial Differential Equations, 24, 47 (2005) · Zbl 1088.35067 · doi:10.1007/s00526-004-0314-5 |
| [11] | P. Esposito, On the existence of blowing-up solutions for a mean field equation,, Ann. Inst. H. Poincaré Anal. Nonlinéaire, 22, 227 (2005) · Zbl 1129.35376 · doi:10.1016/j.anihpc.2004.12.001 |
| [12] | P. Esposito, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent,, J. Diff. Eqns., 227, 29 (2006) · Zbl 1254.35083 · doi:10.1016/j.jde.2006.01.023 |
| [13] | P. Esposito, On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity,, Proc. Lond. math. Soc., 94, 497 (2007) · Zbl 1387.35219 · doi:10.1112/plms/pdl020 |
| [14] | P. Esposito, Concentrating solutions for the Hénon equation in \(\mathbbR^2\),, J. Anal. Math., 100, 249 (2006) · Zbl 1173.35504 · doi:10.1007/BF02916763 |
| [15] | M. Flucher, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots,, Manuscripta Math., 94, 337 (1997) · Zbl 0892.35061 · doi:10.1007/BF02677858 |
| [16] | I. M. Gelfand, Some problems in the theory of quasilinear equations,, Amer. Math. Soc. Transl., 29, 295 (1963) · Zbl 0127.04901 |
| [17] | Z. C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent,, Ann. Inst. H. Poincaré Analyse Non Linéaire, 8, 159 (1991) · Zbl 0729.35014 |
| [18] | D. D. Joseph, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rat. Mech. Anal., 49, 241 (1973) · Zbl 0266.34021 |
| [19] | S. Khenissy, Expansion of the Green’s function for divergence form operators,, C. R. Math. Acad. Sci. Paris, 348, 891 (2010) · Zbl 1198.35061 · doi:10.1016/j.crma.2010.06.024 |
| [20] | L. Ma, Convergence for a Liouville equation,, Comm. Math. Helv., 76, 506 (2001) · Zbl 0987.35056 · doi:10.1007/PL00013216 |
| [21] | K. El Mehdi, Asymptotic estimates and qualitative properties of an elliptic problem in dimension two,, Adv. Nonlineat Stud., 4, 15 (2004) · Zbl 1065.35112 |
| [22] | F. Mignot, Variation d’un point retourment par rapport au domaine,, Comm. Part. Diff. Equations, 4, 1263 (1979) · Zbl 0422.35039 · doi:10.1080/03605307908820128 |
| [23] | X. Ren, On a two-dimensional elliptic problem with large exponent in nonlinearity,, Trans. Amer. Math. Soc., 343, 749 (1994) · Zbl 0804.35042 · doi:10.1090/S0002-9947-1994-1232190-7 |
| [24] | X. Ren, Singular point condensation and least energy solutions,, Proc. Amer. Math. Soc., 124, 111 (1996) · Zbl 0845.35012 · doi:10.1090/S0002-9939-96-03156-5 |
| [25] | O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent,, J. Funct. Anal., 89, 1 (1990) · Zbl 0786.35059 · doi:10.1016/0022-1236(90)90002-3 |
| [26] | O. Rey, A multiplicity result for a variational problem with lack of compactness,, Nonlinear Anal., 13, 1241 (1989) · Zbl 0702.35101 · doi:10.1016/0362-546X(89)90009-6 |
| [27] | J. Wei, Bubbling slutions for an anisotropic Emden-Fowler equation,, Calc. Var. Partial Differential Equations, 28, 217 (2007) · Zbl 1159.35402 · doi:10.1007/s00526-006-0044-y |
| [28] | D. Ye, Une remarque sur le comportement asymptotique des solutions de \(- \Delta u = \lambda f(u)\),, Comp. Rend. Acad. Sci. Paris, 325, 1279 (1997) · Zbl 0895.35014 · doi:10.1016/S0764-4442(97)82353-1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
