Entire solutions of superlinear problems with indefinite weights and Hardy potentials. (English) Zbl 1405.35060
The paper deals with the properties of radial solutions to equations of the form
\[
\Delta u+f(u,|x|)=0,
\]
where \(u: \mathbb{R}^n\to\mathbb{R},\) \(n>2,\) and \(f: \mathbb{R}\times(0,\infty)\) is a \(C^1\) function which is super-linear in \(u\) and \(f(0,|x|)=0.\) By arguing on the corresponding ordinary differential equation, the authors classify the positive and nodal solutions when \(f(u,|x|)\) is negative for \(|x|\) small and positive for \(|x|\) large, or in the opposite situation.
The approach is extended to elliptic equations with Hardy potentials, that is, \[ \Delta u +\dfrac{h(|\mathrm{x}|)}{|{x}|^2} u+ f(u, |{x}|)=0, \] with \(h\) not necessarily constant.
The approach is extended to elliptic equations with Hardy potentials, that is, \[ \Delta u +\dfrac{h(|\mathrm{x}|)}{|{x}|^2} u+ f(u, |{x}|)=0, \] with \(h\) not necessarily constant.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
References:
| [1] | Bae, S, On positive entire solutions of indefinite semilinear elliptic equations, J. Differ. Equ., 247, 1616-1635, (2009) · Zbl 1176.35075 · doi:10.1016/j.jde.2009.06.007 |
| [2] | Bae, S.: Classification of positive solutions of semilinear elliptic equations with Hardy term. Discrete Contin. Dyn. Syst. Supplement(Special issue), 31-39 (2013) · Zbl 1305.35064 |
| [3] | Bamon, R; Pino, M; Flores, I, Ground states of semilinear elliptic equations: a geometric approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, 551-581, (2000) · Zbl 0988.35054 · doi:10.1016/S0294-1449(00)00126-8 |
| [4] | Battelli, F; Johnson, R, On positive solutions of the scalar curvature equation when the curvature has variable sign, Nonlinear Anal., 47, 1029-1037, (2001) · Zbl 1042.34550 · doi:10.1016/S0362-546X(01)00243-7 |
| [5] | Bianchi, G, Non-existence of positive solutions to semilinear elliptic equations on \({\mathbb{R}}^n\) or \({\mathbb{R}}^n_+\) through the method of moving planes, Comm. Partial Differ. Equ., 22, 1671-1690, (1997) · Zbl 0910.35048 · doi:10.1080/03605309708821315 |
| [6] | Bonheure, D; Gomes, J; Habets, P, Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differ. Equ., 214, 36-64, (2005) · Zbl 1210.35089 · doi:10.1016/j.jde.2004.08.009 |
| [7] | Cac, P; Fink, A; Gatica, J, Nonnegative solutions of the radial Laplacian with nonlinearity that changes sign, Proc. Am. Math. Soc., 123, 1393-1398, (1995) · Zbl 0826.34021 · doi:10.1090/S0002-9939-1995-1285979-9 |
| [8] | Calamai, A; Franca, M, Mel’nikov methods and homoclinic orbits in discontinuous systems, J. Dyn. Differ. Equ., 25, 733-764, (2013) · Zbl 1285.34040 · doi:10.1007/s10884-013-9307-4 |
| [9] | Cîrstea, F. C.: A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials. Mem. Amer. Math. Soc. 227(1068), 1-97 (2014) |
| [10] | Coddington, E., Levinson, N.: Theory of Ordinary Differential Equations. Mc Graw Hill, New York (1955) · Zbl 0064.33002 |
| [11] | Coppel, W.A.: Stability and Asymtotic Behavior of Differential Equations. Heath Mathematical Monograph, Lexington (1965) · Zbl 0154.09301 |
| [12] | Coppel, W.A.: Dichotomies in stability theory. Lecture Notes in Mathematics, vol. 377. Springer, Berlin (1978) · Zbl 0376.34001 |
| [13] | Dalbono, F; Franca, M, Nodal solutions for supercritical Laplace equations, Commun. Math. Phys., 347, 875-901, (2016) · Zbl 1433.35132 · doi:10.1007/s00220-015-2546-y |
| [14] | Deng, Y-B; Li, Y; Liu, Y, On the stability of the positive radial steady states for a semilinear Cauchy problem, Nonlinear Anal., 54, 291-318, (2003) · Zbl 1290.35045 · doi:10.1016/S0362-546X(03)00064-6 |
| [15] | Dosla, Z; Marini, M; Matucci, S, Positive solutions of nonlocal continuous second order BVP’s, Dyn. Syst. Appl., 23, 431-446, (2014) · Zbl 1312.34057 |
| [16] | Felli, V; Marchini, E; Terracini, S, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discret. Contin. Dyn. Syst., 21, 91-119, (2008) · Zbl 1141.35362 · doi:10.3934/dcds.2008.21.91 |
| [17] | Felli, V; Marchini, E; Terracini, S, Fountain-like solutions for nonlinear elliptic equations with critical growth and Hardy potential, Commun. Contemp. Math., 7, 867-904, (2005) · Zbl 1199.35083 · doi:10.1142/S0219199705001994 |
| [18] | Ferrero, A; Gazzola, F, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differ. Equ., 177, 494-522, (2001) · Zbl 0997.35017 · doi:10.1006/jdeq.2000.3999 |
| [19] | Fowler, RH, Further studies of emden’s and similar differential equations, Q. J. Math., 2, 259-288, (1931) · JFM 57.0523.02 · doi:10.1093/qmath/os-2.1.259 |
| [20] | Franca, M, Some results on the m-Laplace equations with two growth terms, J. Dyn. Differ. Equ., 17, 391-425, (2005) · Zbl 1097.34020 · doi:10.1007/s10884-005-4572-5 |
| [21] | Franca, M, Non-autonomous quasilinear elliptic equations and wazewski’s principle, Topol. Methods Nonlinear Anal., 23, 213-238, (2004) · Zbl 1075.35013 · doi:10.12775/TMNA.2004.010 |
| [22] | Franca, M, Radial ground states and singular ground states for a spatial dependent \(p\)-Laplace equation, J. Differ. Equ., 248, 2629-2656, (2010) · Zbl 1195.35156 · doi:10.1016/j.jde.2010.02.012 |
| [23] | Franca, M, Positive solutions for semilinear elliptic equations with mixed non-linearities: 2 simple models exhibiting several bifurcations, J. Dyn. Differ. Equ., 23, 573-611, (2011) · Zbl 1254.35087 · doi:10.1007/s10884-010-9198-6 |
| [24] | Franca, M, Positive solutions of semilinear elliptic equations: a dynamical approach, Differ. Integral Equ., 26, 505-554, (2013) · Zbl 1299.35127 |
| [25] | Franca, M; Sfecci, A, On a diffusion model with absorption and production, Nonlinear Anal. Real World Appl., 34, 41-60, (2017) · Zbl 1354.35027 · doi:10.1016/j.nonrwa.2016.07.006 |
| [26] | Gidas, B; Ni, WM; Nirenberg, L, Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^n\), Adv. Math. Suppl. Stud., 7A, 369-402, (1981) · Zbl 0469.35052 |
| [27] | Johnson, R, Concerning a theorem of Sell, J. Differ. Equ., 30, 324-339, (1978) · Zbl 0416.58021 · doi:10.1016/0022-0396(78)90004-9 |
| [28] | Johnson, R; Pan, XB; Yi, YF, Singular ground states of semilinear elliptic equations via invariant manifold theory, Nonlinear Anal., 20, 1279-1302, (1993) · Zbl 0803.35039 · doi:10.1016/0362-546X(93)90132-C |
| [29] | Johnson, R; Pan, XB; Yi, YF, Positive solutions of super-critical elliptic equations and asymptotics, Comm. Partial Differ. Equ., 18, 977-1019, (1993) · Zbl 0793.35029 · doi:10.1080/03605309308820958 |
| [30] | Jones, C; Küpper, T, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17, 803-835, (1986) · Zbl 0606.35032 · doi:10.1137/0517059 |
| [31] | Li, Y; Ni, W, Radial symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}}^n\), Commun. Partial Differ. Equ., 18, 1043-1054, (1993) · Zbl 0788.35042 · doi:10.1080/03605309308820960 |
| [32] | Marini, M; Matucci, S, A boundary value problem on the half-line for superlinear differential equations with changing sign weight, Rend. Istit. Mat. Univ. Trieste, 44, 117-132, (2012) · Zbl 1269.34027 |
| [33] | Matucci, S, A new approach for solving nonlinear BVP’s on the half-line for second order equations and applications, Math. Bohem., 140, 153-169, (2015) · Zbl 1349.34065 |
| [34] | Murray, JD, Mathematical biology. II. spatial models and biomedical applications, (2003), New York · Zbl 1006.92002 |
| [35] | Ni, WM; Serrin, J, Nonexistence theorems for quasilinear partial differential equations, rend, Circ. Mat. Palermo Centen. Suppl. Ser. II, 8, 171-185, (1985) · Zbl 0625.35028 |
| [36] | Papini, D; Zanolin, F, Periodic points and chaotic-like dynamics of planar maps associated to nonlinear hill’s equations with indefinite weight, Georgian Math. J., 9, 339-366, (2002) · Zbl 1015.37027 |
| [37] | Pohozaev, SI, Eigenfunctions of the equations \(Δ u+ λ f(u)=0\), Sov. Math. Dokl, 5, 1408-1411, (1965) · Zbl 0141.30202 |
| [38] | Terracini, S, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differ. Equ., 1, 241-264, (1996) · Zbl 0847.35045 |
| [39] | Wang, X, On the Cauchy problem for reaction-diffusion equations, Trans. Am. Math. Soc., 337, 549-590, (1993) · Zbl 0815.35048 · doi:10.1090/S0002-9947-1993-1153016-5 |
| [40] | Yanagida, E, Structure of radial solutions to \(Δ u +K(|x|) |u|^{p-1}u=0\) in \({\mathbb{R}}^{n}\), SIAM J. Math. Anal., 27, 997-1014, (1996) · Zbl 0856.34040 · doi:10.1137/0527053 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
