On branches of positive solutions for 𝑝-Laplacian problems at the extreme value of the Nehari manifold method

Ilyasov, Yavdat; Silva, Kaye

doi:10.1090/proc/13972

On branches of positive solutions for $p$-Laplacian problems at the extreme value of the Nehari manifold method
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Proc. Amer. Math. Soc. 146 (2018), 2925-2935 Request permission

Abstract:

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the $p$-Laplacian, an indefinite nonlinearity, and depend on a real parameter $\lambda$. A special focus is given to the extreme value $\lambda ^*$ of the Nehari manifold that determines the threshold of applicability of the Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where the parameter $\lambda$ lies above the threshold $\lambda ^*$ is obtained.

References

Stanley Alama and Gabriella Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations 1 (1993), no. 4, 439–475. MR 1383913, DOI 10.1007/BF01206962
Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349–381. MR 0370183, DOI 10.1016/0022-1236(73)90051-7
Henri Berestycki, Italo Capuzzo-Dolcetta, and Louis Nirenberg, Variational methods for indefinite superlinear homogeneous elliptic problems, NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553–572. MR 1356874, DOI 10.1007/BF01210623
Michael G. Crandall and Paul H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis 8 (1971), 321–340. MR 0288640, DOI 10.1016/0022-1236(71)90015-2
Jesús Ildefonso Díaz and José Evaristo Saá, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 12, 521–524 (French, with English summary). MR 916325
Pavel Drábek and Jaroslav Milota, Methods of nonlinear analysis, 2nd ed., Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer Basel AG, Basel, 2013. Applications to differential equations. MR 3025694, DOI 10.1007/978-3-0348-0387-8
Pavel Drábek and Stanislav I. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 4, 703–726. MR 1465416, DOI 10.1017/S0308210500023787
Ya. Sh. Il′yasov, Nonlocal investigations of bifurcations of solutions of nonlinear elliptic equations, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 6, 19–48 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 6, 1103–1130. MR 1970351, DOI 10.1070/IM2002v066n06ABEH000408
Yavdat Il′yasov, On positive solutions of indefinite elliptic equations, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 6, 533–538 (English, with English and French summaries). MR 1860925, DOI 10.1016/S0764-4442(01)01924-3
Yavdat Ilyasov, On extreme values of Nehari manifold method via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017), no. 2, 683–714. MR 3670482, DOI 10.12775/tmna.2017.005
I. Kuzin and S. Pohozaev, Entire solutions of semilinear elliptic equations, Progress in Nonlinear Differential Equations and their Applications, vol. 33, Birkhäuser Verlag, Basel, 1997. MR 1479168
Peter Lindqvist, On the equation $\textrm {div}\,(|\nabla u|^{p-2}\nabla u)+\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), no. 1, 157–164. MR 1007505, DOI 10.1090/S0002-9939-1990-1007505-7
Zeev Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101–123. MR 111898, DOI 10.1090/S0002-9947-1960-0111898-8
Tiancheng Ouyang, On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on compact manifolds. II, Indiana Univ. Math. J. 40 (1991), no. 3, 1083–1141. MR 1129343, DOI 10.1512/iumj.1991.40.40049
S. I. Pokhozhaev, The fibration method for solving nonlinear boundary value problems, Trudy Mat. Inst. Steklov. 192 (1990), 146–163 (Russian). Translated in Proc. Steklov Inst. Math. 1992, no. 3, 157–173; Differential equations and function spaces (Russian). MR 1097896
Patrizia Pucci and Vicenţiu Rădulescu, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Unione Mat. Ital. (9) 3 (2010), no. 3, 543–582. MR 2742781
Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406