Sharp criteria of Liouville type for some nonlinear systems

Abstract

In this paper, we establish the sharp criteria for the nonexistence of positive solutions to the Hardy-Littlewood-Sobolev (HLS) type system of nonlinear equations and the corresponding nonlinear differential systems of Lane-Emden type equations. These nonexistence results, known as Liouville type theorems, are fundamental in PDE theory and applications. A special iteration scheme, a new shooting method and some Pohozaev type identities in integral form as well as in differential form are created. Combining these new techniques with some observations and some critical asymptotic analysis, we establish the sharp criteria of Liouville type for our systems of nonlinear equations. Similar results are also derived for the system of Wolff type integral equations and the system of $\gamma$-Laplace equations. A dichotomy description in terms of existence and nonexistence for solutions with finite energy is also obtained.

References in corpus (1)

• A degree theory approach for the shooting method

Cited by in corpus (6)

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• A degree theory approach for the shooting method
• Decay properties of the Hardy-Littlewood-Sobolev systems of the Lane-Emden type
• A Liouville theorem for the fractional Ginzburg-Landau equation
• Finite energy solutions and critical conditions of nonlinear equations in $R^n$
• Liouville theorems and Fujita exponent for nonlinear space fractional diffusions