Abstract
We are concerned with the parametrized family of problems
where \(\Omega \) is a bounded domain of \({\mathbb {R}}^N~(N\ge 3)\) with regular boundary \(\partial \Omega ,~{\mathcal {L}}\) is a general second-order uniformly elliptic operator, \(\lambda ,~l>0\), \(a:{\overline{\Omega }}\rightarrow {\mathbb {R}}\) is a continuous function which may change sign, \(f:{\mathbb {R}}^+\rightarrow {\mathbb {R}}\) is subcritical and superlinear at infinity. Under some suitable conditions, we obtain there exists \(\lambda _0 > 0\) such that (P) has positive solutions for all \(0 < \lambda \le \lambda _0 \) by topological degree argument and a priori estimates. In doing so, we require f to be of regular variation at infinity.
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References
Allegretto, W., Nistri, P., Zecca, P.: Positive solutions of elliptic non-positone problems. Differ. Integr. Equ. 5(1), 95–101 (1992)
Ambrosetti, A., Arcoya, D., Buffoni, B.: Positive solutions for some semi-positone problems via bifurcation theory. Differ. Integr. Equ. 7(3–4), 655–663 (1994)
Anuradha, V., Hai, D.D., Shivaji, R.: Existence results for superlinear semipositone BVP’s. Proc. Am. Math. Soc. 124(3), 757–763 (1996)
Brezis, H., Turner, R.E.L.: On a class of superlinear elliptic problems. Comm. Part. Differ. Equ. 2, 601–614 (1977)
Berestycki, H., Capuzzo-Dolcetta, I., Nirenberg, L.: Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4(1), 59–78 (1994)
Castro, A., Shivaji, R.: Nonnegative solutions for a class of nonpositone problems. Proc. R. Soc. Edinb. Sect. A 108(3–4), 291–302 (1988)
Castro, A., Shivaji, R.: Non-negative solutions for a class of radially symmetric non-positive problems. Proc. Am. Math. Soc. 106, 735–740 (1989)
Castro, A., Garner, J.B., Shivaji, R.: Existence results for classes of sublinear semipositone problems. Results Math. 23, 214–220 (1993)
Caldwell, S., Castro, A., Shivaji, R., Unsurangsie, S.: Positive solutions for classes of multiparameter elliptic semipositone problems. Electron. J. Differ. Equ. 96, 1–10 (2007)
Costa, D.G., Quoirin, H., Tehrani, H.: A variational approach to superlinear semipositone elliptic problems. Proc. Am. Math. Soc. 145(6), 2661–2675 (2017)
Costa, D.G., Tehrani, H., Yang, J.: On a variational approach to existence and multiplicity results for semipositone problems. Electron. J. Differ Equ. 11, 1–10 (2006)
Drame, A.K., Costa, D.G.: On positive solutions of one-dimensional semipositone equations with nonlinear boundary conditions. Appl. Math. Lett. 25(12), 2411–2416 (2012)
Dancer, E.N., Shi, J.: Uniqueness and nonexistence of positive solutions to semipositone problems. Bull. Lond. Math. Soc. 38(6), 1033–1044 (2006)
García-Melián, J., Iturriaga, L., Ramos Quoirin, H.: A priori bounds and existence of solutions for slightly superlinear elliptic problems. Adv. Nonlinear Stud. 15(4), 923–938 (2015)
Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear elliptic equations. Commun. Part. Differ. Equ. 6(8), 883–901 (1981)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, NY (1983)
Kaufmann, U., Ramos Quoirin, H.: Positive solutions of indefinite semipositone problems via sub-super solutions. Differ. Integr. Equ. 31(7/8), 497–506 (2018)
Seneta, E.: Regularly Varying Functions. Lecture Notes in Mathematics, vol. 508. Springer, Berlin (1976)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. 1. Springer, Berlin (1985)
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The authors are very grateful to the anonymous referees for their valuable suggestions.
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This work was supported by National Natural Science Foundation of China (No. 12061064) and Shanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSY018).
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Y. Zhang: Supported by the NSFC (No. 12061064) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (No. 22JSY018).
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Ma, R., Zhang, Y. & Zhu, Y. Positive Solutions of Indefinite Semipositone Elliptic Problems. Qual. Theory Dyn. Syst. 23, 45 (2024). https://doi.org/10.1007/s12346-023-00901-0
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DOI: https://doi.org/10.1007/s12346-023-00901-0