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Existence of ground states for quasilinear nonhomogeneous elliptic systems. (English) Zbl 0965.35040

The authors prove the existence of positive radial solutions (decaying to zero as \(|x|\to\infty\)) of the quasilinear system \(-\text{div} (A_i(|\nabla u_i|)\nabla u_i)=\sum_j a_{ij}(|x|)f_{ij}(u_j)\), \(x\in\mathbb R^N\), \(N>1\), \(i=1,2,\dots,n\). The functions \(A_i,a_{ij},f_{ij}\) are continuous, \(A_i>0\), \(a_{ij}\geq 0\), \(f_{ij}(0)=0\), \(f_{ij}(s)>0\) for \(s>0\), \(f_{ij}(s)\to\infty\) as \(s\to\infty\). Moreover, the functions \(A_i,f_{ij}\) are supposed to be asymptotically homogeneous at zero and at infinity (with suitable powers satisfying several inequalities) and an integrability condition involving \(A_i,a_{ij}\) has to be satisfied. The proof is based on the use of the Leray-Schauder degree and on a modification of a priori estimates of B. Gidas and J. Spruck [Commun. Partial Differ. Equations 6, 883-901 (1981; Zbl 0462.35041)]. Two examples with nonlinearities \(A_i,f_{ij}\) of the form \(s^p\log(1+s)\), \(s^p\) and \(\log(1+s^p)/\log(1+s^{-q})\) are given.

MSC:

35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35B45 A priori estimates in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 0462.35041
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References:

[1] Beresticky, H.; Lions, P. L.; Peletier, L. A., An O.D.E. approach to the existence of positive solutions of semilinear problems in \(\textbf{R}^N \), Indiana Univ. Math. J., 30, 141-157 (1981)
[2] G. Caristi, and, E. Mitidieri, Non-existence of positive solutions of quasilinear differential inequalities, Proceedings of the conference “Differential Equations, Ferrara, 1996”. [In Italian]; G. Caristi, and, E. Mitidieri, Non-existence of positive solutions of quasilinear differential inequalities, Proceedings of the conference “Differential Equations, Ferrara, 1996”. [In Italian] · Zbl 0887.35063
[3] Caristi, G.; Mitidieri, E., Non-existence of positive solutions of quasilinear equations, Adv. Differential Equations, 2, 319-359 (1997) · Zbl 1023.34500
[4] Clément, P.; Manasevich, R.; Mitidieri, E., Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18, 2071-2106 (1993) · Zbl 0802.35044
[5] Dambrosio, W., Multiple solutions of weakly-coupled systems with \(p\)-laplacian operators, Results in Math., 36, 34-54 (1999) · Zbl 0942.34015
[6] Filippucci, R.; Pucci, P., Non-existence and other properties of quasilinear elliptic equations, Differential and Integral Equations, 8, 325-538 (1995) · Zbl 0815.35004
[7] Franchi, B.; Lanconelli, E.; Serrin, J., Existence and uniqueness of nonnegative solutions of quasilinear equations in \(R^n\), Adv. in Math., 118, 177-243 (1996) · Zbl 0853.35035
[8] Gidas, B.; Spruck, J., A priori bounds for positive solutions of nonlinear elliptic equations, Comm. in Partial Differential Equations, 6, 883-901 (1981) · Zbl 0462.35041
[9] Garcı́a-Huidobro, M.; Guerra, I.; Manasevı́ch, R., Existence of positive radial solutions for a weakly coupled system via blow up, Abstract Appl. Anal., 3, 105-131 (1998) · Zbl 0965.35058
[10] Garcı́a-Huidobro, M.; Manasevich, R.; Schmitt, K., Some bifurcation results for a class of \(p\)-Laplacian like operators, Differential and Integral Equations, 10, 51-66 (1997) · Zbl 0879.34029
[11] Garcı́a-Huidobro, M.; Manasevich, R.; Ubilla, P., Existence of positive solutions for some Dirichlet problems with an asymptotically homogeneous operator, Electron J. Differential Equations, 10, 1-22 (1995) · Zbl 0823.35057
[12] Kusano, T.; Naito, M., Positive radial solutions of superlinear elliptic equations, Hiroshima Math. J., 16, 361-366 (1986) · Zbl 0611.35021
[13] Mitidieri, E., A Rellich-type identity and applications, Comm. Partial Differential Equations, 18, 125-151 (1993) · Zbl 0816.35027
[14] Mitidieri, E., Non-existence of positive solutions of semilinear elliptic systems in \(R^N\), Differential Integral Equations, 9 (1995)
[15] Mitidieri, E.; Sweers, G.; van der Vorst, R. C.A. M., Non-existence theorems for systems of quasilinear partial differential equations, Differential Integral Equations, 8, 1331-1354 (1995) · Zbl 0833.35043
[16] Ni, W.-M.; Serrin, J., Existence and non-existence theorems for ground states for quasilinear partial differential equations, The anomalous case, Atti Convegni Lincei, 77, 231-257 (1985)
[17] Ni, W.-M.; Serrin, J., Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, 8, 171-185 (1985) · Zbl 0625.35028
[18] Ni, W.-M.; Serrin, J., Existence and non-existence theorems for ground states for quasilinear partial differential equations, Atti Convegni Lincei, 77, 231-257 (1985)
[19] Serrin, J.; Zou, H., Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications 3 (1994), Texas A&M UniversityDepartment of Mathematics: Texas A&M UniversityDepartment of Mathematics College Station, p. 55-68 · Zbl 0900.35121
[20] Serrin, J.; Zou, H., Non-existence of positive solutions of the Lame-Emdem system, Differential and Integral Equations, 9, 635-653 (1996) · Zbl 0868.35032
[21] Serrin, J.; Zou, H., Existence of positive entire solutions of elliptic Hamiltonian systems, Comm. and Partial Differential Equations, 23, 577-599 (1998) · Zbl 0906.35033
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