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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 \(a(x)\).

MSC:
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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