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Hartman-type results for $$p(t)$$-Laplacian systems. (English) Zbl 1025.34017
Summary: Consider the weighted $$p(t)$$-Laplacian ordinary system $-\biggl(w(t) \bigl|u'(t)\bigr |^{p(t)-2}u'(t)\biggr)'+ w(t)f\bigl(t, u(t) \bigr)=0\text{ in }(a,b),\;u(a)= u(b)=0,$ with $$f\in C([a,b]\times \mathbb{R}^N, \mathbb{R}^N)$$, $$w\in C([a,b], \mathbb{R})$$, $$p\in C([a,b],\mathbb{R})$$ and $$p(t)>1$$ for $$t\in [a,b]$$. It is proved that if $$\exists R>0$$ such that $$\langle f(t,u),u\rangle\geq 0$$, $$\forall t\in[a,b]$$, $$\forall u\in\mathbb{R}^N$$ with $$|u|=R$$, then the problem has a solution $$u$$ such that $$|u(t)|\leq R$$ for $$t\in[a,b]$$. As a corollary of this result, taken $$w(t)=t^{n-1}$$, the authors obtain the existence of the radial solutions for elliptic systems. This result generalizes the corresponding results obtained by Hartman and Mawhin.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
##### Keywords:
 [1] Fan, X.L., p(x)-Laplace operator, J. gansu education college (NS), 13, 1, 1-5, (1999) [2] Fan, X.L.; Zhao, D., The quasi-minimizer of integral functionals with m(x) growth conditions, Nonlinear anal., 39, 807-816, (2000) · Zbl 0943.49029 [3] Fan, X.L.; Zhao, D., Local C1,α regularity of weak solutions for p(x)-Laplacian equations, J. gansu education college (NS), 15, 2, 1-5, (2001) [4] Fan, X.L.; Zhao, D., On the spaces $$L\^{}\{p(x)\}(Ω)$$ and $$W\^{}\{m,p(x)\}(Ω)$$, J. math. anal. appl., 262, 749-760, (2001) · Zbl 0995.46023 [5] Fan, X.L.; Zhao, Y.Z.; Zhao, D., Compact imbedding theorems with symmetry of strauss – lions type for the space $$W\^{}\{1,p(x)\}(Ω)$$, J. math. anal. appl., 255, 333-348, (2001) · Zbl 0988.46025 [6] Giaqunta, M., Multiple integrals in the calculus of variations and nonlinear elliptic systems, (1983), Princeton University Press Princeton, NJ [7] Hartman, P., On boundary value problems for systems of ordinary nonlinear second order differential equations, Trans. amer. math. soc., 96, 493-509, (1960) · Zbl 0098.06101 [8] Heinonen, J.; Kilpelainen, T.; Martio, O., Nonlinear potential theory of degenerate elliptic equations, (1993), Clarendon Press Oxford · Zbl 0780.31001 [9] Mawhin, J., Some boundary value problems for hartman-type perturbations of the ordinary vector p-Laplacian, Nonlinear anal., 40, 497-503, (2000) · Zbl 0959.34014 [10] Musielak, J., Orlicz spaces and modular spaces, Lecture notes in mathematics, Vol. 1034, (1983), Springer Berlin · Zbl 0557.46020 [11] Palais, R.S., The principle of symmetric criticality, Comm. math. phys., 69, 19-30, (1979) · Zbl 0417.58007 [12] Zeidler, E., Nonlinear functional analysis and its applications, II/B: nonlinear monotone operators, (1990), Springer New York [13] Zhao, D., The local regularity of weak solutions of elliptic equations of divergence form with p(x)-growth conditions, doctoral dissertation, (1998), Lanzhou University Lanzhou, China [14] Zhikov, V.V., Averaging of functions in the calculus of variation and elasticity theory, Math. USSR-izv., 29, 33-36, (1987) · Zbl 0599.49031 [15] Zhikov, V.V., Weighted Sobolev spaces, Sbornik: mathematics, 189, 8, 1139-1170, (1998) · Zbl 0919.46026