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Existence and structure of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems. (English) Zbl 1142.35509
This paper is devoted to the study of the existence of positive solutions of the sublinear weighted elliptic mixed boundary value problem \[ \begin{gathered} Lu=\lambda W(x)u- a(x) f(x,u)u\quad\text{in }\Omega,\\ {\mathcal B}(b)u= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(a(x)\in L^\infty(\Omega)\), \(a(x)\geq 0\) and \(W\in L^\infty(\Omega)\). The author analyses the structure of the diagram of positive solutions of (1), accordingly to the sign of potential \(W\) in the vanishing set of \(a(x)\).

35L65 Hyperbolic conservation laws
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
47J15 Abstract bifurcation theory involving nonlinear operators
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[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Commun. Pure Appl. Math. XII (1959) 623-727. · Zbl 0093.10401
[2] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044
[3] Amann, H., Dual semigroups and second order linear elliptic boundary value problems, Israel J. math., 45, 225-254, (1983) · Zbl 0535.35017
[4] Amann, H.; López-Gómez, J., A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. differential equations, 146, 336-374, (1998) · Zbl 0909.35044
[5] Brezis, H.; Oswald, L., Remarks on sublinear elliptic equations, Nonlinear anal. TMA, 10, 55-64, (1986) · Zbl 0593.35045
[6] S. Cano-Casanova, J. López-Gómez, Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, 1999, Preprint.
[7] Crandall, M.G.; Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. rational mech. anal., 67, 161-180, (1973) · Zbl 0275.47044
[8] Fraile, J.M.; Koch-Medina, P.; López-Gómez, J.; Merino, S., Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. differential equations, 127, 295-319, (1996) · Zbl 0860.35085
[9] Garcı́a-Melián, J.; Gómez-Reñasco, R.; López-Gómez, J.; Sabina de Lis, J.C., Pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs, Arch. rational mech. anal., 145, 261-289, (1998) · Zbl 0926.35036
[10] J. López-Gómez, Bifurcation theory, Notas del curso de Zurich, 1994.
[11] López-Gómez, J., The maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems, J. differential equations, 127, 263-294, (1996) · Zbl 0853.35078
[12] López-Gómez, J., Large solutions, metasolutions and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems with refuges, Elec. J. differential equations, 5, 135-171, (2000) · Zbl 1055.35049
[13] López-Gómez, J.; Molina-Meyer, M., The maximum principle for cooperative weakly coupled elliptic systems and some applications, Differential int. equations, 7, 383-398, (1994) · Zbl 0827.35019
[14] López-Gómez, J.; Sabina de Lis, J.C., First variations of principal eigenvalues with respect to the domain and point-wise growth of positive solutions for problems where bifurcation from infinity occurs, J. differential equations, 148, 47-64, (1998) · Zbl 0915.35080
[15] Murray, J.D., Mathematical biology vol. 18, Springer biomathematics texts, (1993), Springer Berlin
[16] Necas, J., LES méthodes directes en théorie des équations elliptiques, (1967), Academia Prague
[17] Okubo, A., Diffusion and ecological problems: mathematical models, (1980), Springer New York · Zbl 0422.92025
[18] Ouyang, T., On the positive solutions of semilinear equations δ u+λu−hup=0 on the compact manifolds, Trans. amer. math. soc., 331, 503-527, (1992)
[19] Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. funct. anal., 7, 487-513, (1971) · Zbl 0212.16504
[20] Schaefer, H.H., Topological vector spaces, (1971), Springer New York · Zbl 0212.14001
[21] Stein, E.M., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton, NJ · Zbl 0207.13501
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