Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity. (English) Zbl 0785.35045

Summary: For the Emden-Fowler equation \(-\Delta u=\lambda e^ u\) in \(\Omega\subset\mathbb{R}^ 2\), the connectivity of the trivial solution and the one-point blow-up singular limit is studied with respect to the parameter \(\lambda>0\). The connectivity is assured when the domain \(\Omega\) is simply connected and the total mass \(\sum=\int_ \Omega\lambda e^ udx\) tends to \(8\pi\) from below, which is a generalization for the case that \(\Omega\) is a ball.


35J65 Nonlinear boundary value problems for linear elliptic equations
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J60 Nonlinear elliptic equations
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[1] Bandle, C., Existence Theorems, Qualitative Results and a priori Bounds for a Class of Nonlinear Dirichlet Problems, Arch. Rat. Mech. Anal., Vol. 58, 219-238 (1975) · Zbl 0335.35046
[2] Bandle, C., On a Differential Inequality and its Application to Geometry, Math. Z., Vol. 147, 253-261 (1976) · Zbl 0316.35009
[3] Bandle, C., Isoperimetric Inequalities and Applications (1980), Pitman: Pitman Boston-London-Melbourne · Zbl 0436.35063
[4] Cheng, S. Y., Eigenfunctions and Nodal Sets, Comment. Math. Helvetici, Vol. 51, 43-55 (1976) · Zbl 0334.35022
[5] Crandall, M. G.; Rabinowitz, P. H., Some Continuation and Variational Methods for Positive Solutions of Nonlinear Elliptic Eigenvalu Problems, Arch. Rat. Mech. Anam., Vol. 58, 207-218 (1975) · Zbl 0309.35057
[6] Fujita, H., On the Nonlinear Equations Δ \(u + e^u = 0\) and ∂\(v\)/∂\(t = Δ v + e^v\), Bull. Am. Math. Soc., Vol. 75, 132-135 (1969) · Zbl 0216.12101
[7] Kato, T., Perturbation Theory for Linear Operators (1976), Springer: Springer Berlin-Heidelberg-New York
[8] Keener, J. P.; Keller, H. B., Positive Solutions of Convex Nonlinear Eigenvalue Problem, J. Diff. Eq., Vol. 16, 103-125 (1974) · Zbl 0287.35074
[9] Keller, H. B.; Cohen, D. S., Some Positive Problems Suggested by Nonlinear Heat Generation, J. Math. Mech., 16, 1361-1376 (1967) · Zbl 0152.10401
[10] Laetsch, T., On the Number of Boundary Value Problems with Convex Nonlinearities, J. Math. Anal. Appl., Vol. 35, 389-404 (1971) · Zbl 0191.40102
[11] Lin, S. S., On Non-Radially Symmetric Bifurcations in the Annulus, J. Diff. Eq., 80, 251-279 (1989) · Zbl 0688.35005
[12] Liouville, J., Sur l’équation aux différences partielles ∂\(^2\) Log λ/∂\(u\) ∂\(v\) ± λ/\(2 a^2 = 0\), J. Math., Vol. 18, 71-72 (1853)
[13] Moseley, J. L., A Two-Dimensional Dirichlet Problem with an Exponential Nonlinearily, SIAM J. Math. Anal., Vol. 14, 934-946 (1983) · Zbl 0543.35036
[14] Nagasaki, K.; Suzuki, T., Radial and Nonradial Solutions for the Nonlinear Eigenvalue problem Δ \(u\) + λ \(e^u = 0\) on Annului in \(R^2\), J. Diff. Eq., Vol. 87, 144-168 (1990) · Zbl 0717.35030
[15] Nagasaki, K.; Suzuki, T., Asymptotic Analysis for Two-Dimensional Elliptic Eigenvalue Problems with Exponentially-Dominated Nonlinearities, Asymptotic Analysis, Vol. 3, 173-188 (1990) · Zbl 0726.35011
[16] Nehari, Z., On the Principal Frequency of a Membrane, Pac. J. Math., Vol. 8, 285-293 (1958) · Zbl 0086.19204
[17] Pleijel, Å., Remarks on Courant’s Nodal Line Theorem, Comm. Pure Appl. Math., Vol. 9, 543-550 (1956) · Zbl 0070.32604
[18] Suzuki, T.; Nagasaki, K., On the Nonlinear Eigenvalue Problem Δ \(u\) + λ \(e^u = 0\), Trans. Am. Math. Soc., Vol. 309, 591-608 (1988) · Zbl 0711.35043
[19] Wente, H., Counterexample to a Conjecture of H. Hopf, Pacific J. Math., Vol. 121, 193-243 (1986) · Zbl 0586.53003
[20] Weston, V. H., On the Asymptotic Solution of a Partial Differential Equation with an Exponential Nonlinearity, SIAM J. Math. Anal., Vol. 9, 1030-1053 (1978) · Zbl 0402.35038
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