Abstract
The present paper deals with the study of semilinear and non-homogeneous Schrödinger equations on a manifold with conical singularity. We provide a suitable constant by Sobolev embedding constant and for p ∈ (2, 2∗) with respect to non-homogeneous term g(x) ∈ L n/22 (B), which helps to find multiple solutions of our problem. More precisely, we prove the existence of two solutions to the problem 1.1 with negative and positive energy in cone Sobolev space H 1,n/22,0 (B). Finally, we consider p = 2 and we prove the existence and uniqueness of Fuchsian-Poisson problem.
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M. Alimohammady and M. K. Kalleji, Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration, J. Funct. Anal., 265 (2013), 2331–2356.
H. Chen, X. Liu and Y. Wei, Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on manifold with conical singularities, Calc. Var., 43 (2012), 463–484.
H. Chen, X. Liu and Y. Wei, Existence theory for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents, Ann. Glob. Anal. Geom., 37 (2011), 27–43.
H. Chen, Y. Wei and B. Zhou, Existence of solutions for degenerate elliptic equations with singular potential on conical singular manifolds, Math. Nachr., 25 (No.11-12) (2012), 1370–1384.
J. Dávila and I. Perla, Nonlinear elliptic problems with a singular weight on the boundary, Calc. Var. Partial Differential Equations, 41(3-4) (2011), 567–586.
M. Badiale and E. Serra, Semilinear elliptic equation for beginners, DOI: 10.1007/978-0-85729-227-8, Springer London, 20 (2011).
Y. Jiang, Z. Wang and H. S. Zhou, Multiple solution for a nonhomogeneous Schrödinger-Maöxwell system in R3, Nonlinear Anal., 83 (2013), 50–57.
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazertype problem on RN, Proc. Roy. Soc. Edinburgh Sect. A, 129(4), (2004), 787–809.
B. W. Schulze, Boundary value problems and singular pseudo-differential operators, Wiley, Chichester, (1998).
E. Schrohe and J. Seiler, Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces, Integr. Equ. Oper. Theory, 41 (2001), 93–114.
X. P. Zhu and H. S. Zhou, Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 115(2), (1990), 301–318.
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Alimohammady, M., Jafari, A.A. & Kalleji, M.K. Multiple solutions for non-homogeneous degenerate Schrödinger equations in cone Sobolev spaces. Indian J Pure Appl Math 48, 133–146 (2017). https://doi.org/10.1007/s13226-017-0215-x
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DOI: https://doi.org/10.1007/s13226-017-0215-x