# zbMATH — the first resource for mathematics

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
Full Text:
##### 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 RN, 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] · 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 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.