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Non existence of bounded-energy solutions for some semilinear elliptic equations with a large parameter. (English) Zbl 1121.35053
The authors prove that the energy of positive solutions to the Dirichlet problem \[ \begin{cases} -\Delta u+\lambda u=u^{{N+2}\over{N-2}} & \text{ in}\;\Omega\subset \mathbb R^N, \quad u>0 \text{ in } \Omega,\cr u=0 & \text{ on}\;\partial\Omega, \end{cases} \] where \(\Omega\) is a smooth bounded domain, \(N\geq 3\), tends to infinity as \(\lambda\to+\infty.\) Moreover it is proved that the bounded energy solutions of mixed (Dirichlet-Neumann) boundary value problems have at least one blow-up point on the Neumann component as \(\lambda\to+\infty.\)

MSC:
35J60 Nonlinear elliptic equations
35B33 Critical exponents in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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[1] ADIMURTHI - G. MANCINI, The Neumann problem for elliptic equations with critical non-linearity, A tribute in honor to G. Prodi, Scuola Normale Superiore di Pisa, 1991, pp. 9-25. Zbl0836.35048 · Zbl 0836.35048
[2] ADIMURTHI - G. MANCINI - S. L. YADAVA, The role of the mean curvature in semilinear Neumann problem involving critical exponent, Communications in Partial Differential Equations, 20 no. 3/4 (1995), pp. 591-631. Zbl0847.35047 MR1318082 · Zbl 0847.35047 · doi:10.1080/03605309508821110
[3] ADIMURTHI - F. PACELLA - S. L. YADAVA, Characterization of concentration points and LQ estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Differential and Integral Equations, 8, no. 1 (1995), pp. 41-68. Zbl0814.35029 MR1296109 · Zbl 0814.35029
[4] D. CAO - E. S. NOUSSAIR - S. YAN, Existence and nonexistence of interior peaked solutions for a nonlinear Neumann problem, Pacific Journal of Mathematics, 200, no. 1 (2001), pp. 19-41. Zbl1140.35440 MR1863405 · Zbl 1140.35440 · doi:10.2140/pjm.2001.200.19
[5] P. CHERRIER, Problems de Neumann nonlineaires sur les varietes riemanniennes, Journal of Functional Analysis, 57 (1984), pp. 154-206. Zbl0552.58032 MR749522 · Zbl 0552.58032 · doi:10.1016/0022-1236(84)90094-6
[6] O. DRUET - E. HEBEY - M. VAUGON, Pohozaev type obstructions and solutions of bounded energy for quasilinear elliptic equations with critical Sobolev growth. The conformally flat case, Nonlinear Anal., 51, no. 1 (2002), Ser A: Theory Methods, pp. 79-94. Zbl1066.35032 MR1915742 · Zbl 1066.35032 · doi:10.1016/S0362-546X(01)00813-6
[7] V. FELLI - E. HEBEY - F. ROBERT, Fourth order equations of critical Sobolev growth. Energy function and solutions of bounded energy in the conformally flat case, preprint, June 2002. Zbl1086.58009 MR2184079 · Zbl 1086.58009 · doi:10.1007/s00030-005-0011-x
[8] N. GHOUSSUB - C. GUI - M. ZHU, On a singularly perturbed Neumann problem with the critical exponent, Comm. P.D.E., 6, no. 11-12 (2001), pp. 1929-1946. Zbl0997.35021 MR1876408 · Zbl 0997.35021 · doi:10.1081/PDE-100107812
[9] D. GILBARG - N. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer, 1983. Zbl0562.35001 MR737190 · Zbl 0562.35001
[10] E. HEBEY, Nonlinear elliptic equations of critical Sobolev growth from a dynamical point of view, preprint. · Zbl 1210.35092
[11] G. MANCINI - R. MUSINA, The role of the boundary in some semilinear Neumann problems, Rendiconti del Seminario Matematico della Universit√† di Padova, 88 (1992), pp. 127-138. Zbl0814.35037 MR1209119 · Zbl 0814.35037 · numdam:RSMUP_1992__88__127_0 · eudml:108263
[12] O. REY, The question of interior blow-up points for an elliptic problem: the critical case, Journal de Mathematiques pures et appliquees, 81 (2002), pp. 655-696. Zbl1066.35033 MR1968337 · Zbl 1066.35033 · doi:10.1016/S0021-7824(01)01251-X
[13] O. REY, Boundary effect for an elliptic Neumann problem with critical nonlinearity, Communications in partial differential equations, 22, no. 7-8 (1997), pp. 1055-1139. Zbl0891.35040 MR1466311 · Zbl 0891.35040 · doi:10.1080/03605309708821295
[14] M. STRUWE, Variational Methods, applications to nonlinear partial differential equations and Hamiltonian systems, Springer, 1990. Zbl0746.49010 MR1078018 · Zbl 0746.49010
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