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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.

##### MSC:
 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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##### References:
 [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 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 δu + e^u = 0 and ∂v/∂t = δv + e^v, Bull. Am. Math. Soc., Vol. 75, 132-135, (1969) · Zbl 0216.12101 [7] Kato, T., Perturbation theory for linear operators, (1976), 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 δu + λe^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 δu + λe^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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