# zbMATH — the first resource for mathematics

Existence of positive solutions for semilinear elliptic equations in general domains. (English) Zbl 0664.35029
The authors consider the existence of positive solutions to the Dirichlet problem $(1)\quad \Delta u(x)+f(u(x))=0,\quad x\in \Omega;\quad u(x)=0,\quad x\in \partial \Omega,$ where $$\Omega$$ is a bounded domain in $$R^ n$$ with smooth boundary and f is a continuous function on R. For it, they introduce the notion of “eccentricity”, e($$\Omega)$$, of a domain $$\Omega$$ with the property that $$1\leq e(\Omega)<+\infty$$ and $$e(\Omega)=1$$ if and only if $$\Omega$$ is a ball, and the notion of the “nonlinearity” of f, N(f) (for instance, if f is a linear function, $$N(f)=1$$; if $$f(u)=u^ k$$, $$k<1$$, $$N(f)=\infty$$ and if $$f(u)=u^ k$$, $$k>1$$, $$N(f)<1)$$. They prove, using a variation of the method of upper and lower solutions, that if $$N(f)>e(\Omega)$$ then there exist positive solutions to (1) on all domains $$\lambda$$ $$\Omega$$ if $$\lambda$$ is sufficiently large (equivalently, positive solutions to the Dirichlet problem for $$\Delta u+\mu f(u)=0$$ exist on $$\Omega$$ for some range of $$\mu)$$. Then, they give some applications of this result.
Also, they consider in more detail the case where $$\Omega$$ is an n-ball and the Neumann problem on n-balls.
The used method may be applied to more general elliptic operators.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J25 Boundary value problems for second-order elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Full Text:
##### References:
 [1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in Banach spaces, SIAM Rev., 18, 620–709 (1976). · Zbl 0345.47044 · doi:10.1137/1018114 [2] Berestycki, H., & P. L. Lions, Existence of stationary states in nonlinear scalar field equations, in Bifurcation Phenomena in Mathematical Physics and Related Topics, ed. by C. Bardos & D. Bessis, D. Reidel Publ. Co.: Doredrecht, Boston, London, (1980). [3] Brezis, H., & L. Swald, Remarks on sublinear elliptic equations, (preprint). [4] Gidas, B., Ni, W. M., & L. Nirenberg, Symmetry of positive solutions of nonlinear e · Zbl 0425.35020 · doi:10.1007/BF01221125 [5] Hess, P., On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. P.D.E., 6, 951–961 (1981). · Zbl 0468.35073 · doi:10.1080/03605308108820200 [6] Hirsch, M. W., Differential Topology, Springer-Verlag: Berlin, Heidelberg, New York, (1976). [7] Lions, P. L., On the existence of positive solutions of semilinear elliptic equations, SIAM, Rev. 24, 441–467 (1982). · Zbl 0511.35033 · doi:10.1137/1024101 [8] Nussbaum, R., Positive solutions of nonlinear elliptic boundary-value problems, J. Math. Anal. Appl. 51, 461–482 (1975). · Zbl 0304.35047 · doi:10.1016/0022-247X(75)90133-X [9] Pohozaev, S. I., Eigenfunctions of the equation 249-01, Sov. Math. Dok. 5, 1408–1411 (1965). [10] Smoller, J., & A. Wasserman, Existence, uniqueness, and nondegeneracy of positive solutions of semilinear elliptic equations, Comm. Math. Phys. 95, 129–159, (1984). · Zbl 0582.35046 · doi:10.1007/BF01468138 [11] Smoller, J., & A. Wasserman, Symmetry-breaking for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal. 95, 217–225 (1986). · Zbl 0629.35040 · doi:10.1007/BF00251359 [12] Smoller, J., & A. Wasserman, Symmetry-breaking for semilinear elliptic equations with general boundary conditions, Comm. Math. Phys. 105, 415–441 (1986). · Zbl 0608.35004 · doi:10.1007/BF01205935 [13] Smoller, J., & A. Wasserman, An existence theorem for positive solutions of semilinear elliptic equations, Arch. Rational Mech. Anal. 95, 211–216 (1986). · Zbl 0624.35034 · doi:10.1007/BF00251358 [14] Smoller, J., Shock Waves and Reaction-Diffusion Equations, Springer-Verlag: Berlin, Heidelberg, New York, (1983). · Zbl 0508.35002
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.