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Sublinear elliptic equations in \(\mathbb{R}{}^ n\). (English) Zbl 0761.35027

The main result is \(-\Delta u=\rho(x)u^ \alpha\), \(0<\alpha<1\), \(\rho\geq 0\), \(0\not\equiv\rho\in L^ \infty_{loc}\) has a bounded solution in \(\mathbb{R}^ n\), \(n\geq 3\), if and only if \(\rho\) satisfies property \((H)\), i.e. \(\Delta u=\rho\) in \(\mathbb{R}^ n\) has a bounded solution. Moreover there is a minimal positive solution of (1). Uniqueness results are established.
Reviewer: J.Frehse (Bonn)

MSC:

35J60 Nonlinear elliptic equations
35B35 Stability in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B50 Maximum principles in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] H. Brezis–L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Analysis,10 (1986), p. 55–64 · Zbl 0593.35045 · doi:10.1016/0362-546X(86)90011-8
[2] E. DiBenedetto, Continuity of weak solutions to a general porous medium equation, Indiana University, Math. J.32 (1983), p. 83–118 · Zbl 0526.35042 · doi:10.1512/iumj.1983.32.32008
[3] A.L. Edelson, Asymptotic properties of semilinear equations, Can. Math. Bull32 (1989), p. 34–46 · Zbl 0634.34019 · doi:10.4153/CMB-1989-006-3
[4] H. Egnell, Asymptotic results for finite energy solutions of semilinear elliptic equations (to appear) · Zbl 0778.35009
[5] E. Eidus, The Cauchy problem for the nonlinear filtration equation in an inhomogeneous medium, J. Diff. Eq.84 (1990), p. 309–318 · Zbl 0707.35074 · doi:10.1016/0022-0396(90)90081-Y
[6] D. Eidus–S. Kamin, in preparation
[7] P. Hess, On the uniqueness of positive solutions of nonlinear elliptic boundary value problems, Math. Z.154 (1977), p. 17–18 · Zbl 0352.35046 · doi:10.1007/BF01215108
[8] S. Kamin–P. Rosenau, Nonlinear thermal evolution in an inhomogeneous medium, J. Math. Phys.23 (1982), p. 1385–1390. · Zbl 0499.76111 · doi:10.1063/1.525506
[9] T. Kato, Schrödinger operators with singular potentials, Israel J. Math. 13 (1972), p. 135–148 · Zbl 0246.35025 · doi:10.1007/BF02760233
[10] M. Krasnoselskii,Positive solutions of operator equations, Noordhoff (1964)
[11] M. Naito, A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J. 14 (1984), p. 211–214 · Zbl 0555.35044
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