On some nonlinear elliptic systems

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Abstract

We study a class of nonvariational elliptic systems involving the p-Laplacian. Under sublinear and superlinear assumptions on the nonlinearities, we show the existence of solutions using fixed point theorems.

Introduction

The aim of this paper is to prove some existence results for nonlinear and nonvariational elliptic systems of the form (S)-Δpu=λf(x,u,v)in RN,-Δqv=μg(x,u,v)in RN,where Δp is the so-called p-Laplacian operator i.e. Δpu=div(|u|p-2u). f and g are real-valued functions with domain RN×R×R; u and v are unknown real-valued functions defined in RN and belonging to appropriate function spaces; λ and μ are positive parameters; p and q are real numbers satisfying 2p,q<N.

Problems involving the “p-Laplacian” arise from many branches of pure mathematics as in the theory of quasiregular and quasiconformal mappings (see [17]) as well as from various problems in mathematical physics notably the flow of non-Newtonian fluids. Roughly speaking, there are two distinct cases: the situation p>2 corresponds to dilatant fluids while the situation 1<p<2 describes pseudo-plastic fluids (see [1], [2]). Observe that the case p=2 of course yields the Laplacian operator which expresses Newtonian fluids.

In recent years, several authors use different methods to solve quasilinear equations or systems defined in bounded or unbounded domains. In the scalar case, weak solutions are obtained generally as critical points of a suitable energy function. There is a large literature on this topic (see [11], [18], etc.). This variational approach is also employed to deal with systems deriving from a potential, i.e. the nonlinearities on the right-hand side are the gradient of a C1-functional satisfying some growth conditions. In this connection, we have considered the mixed case in paper [10], (see also [4], [8]). At the same time, when p=q=2;f=H/v;g=H/u (i.e. H is a C1-Hamiltonian), the system (S) has again a variational structure and the existence of solutions can be proved exploiting critical point theory in an appropriate interpolation space. However, the situation is quite different for nonvariational systems. The more successful method to obtain solutions is based on an inspection of the fixed point theorems. Nevertheless, a method combining both a priori bounds and Leray–Schauder degree was investigated in [7] in order to show the existence of positive radial solutions of quasilinear systems defined in bounded domains. To get a priori estimates, the authors have extended the so called “blow-up” method introduced by Gidas and Spruck in their fundamental paper [12] for semilinear equations. The “blow-up” method is also used in [5], [6], to study sublinear systems defined in bounded domains. Conversely, it seems that this method fails for strong nonlinearities as it was pointed out in [6].

In this work, we treat system (S), first when f and g verify sublinear growth conditions. We show an existence result using a fixed point theorem due to Bohnenblust–Karlin. So, we give a different approach for this kind of systems. The next step, we examine the superlinear case for restricted class of nonlinearities. We establish existence of nontrivial positive radial solutions vanishing at infinity, namely “ground states”, related to a fixed point theorem developed in [3], [13]. Notice that in the first case, λ and μ can be taken equal to 1, in the second one they are supposed to be small enough.

This paper is organized as follows. In Section 2, we establish an existence theorem under sublinear growth conditions. Whereas, Section 3 is devoted to deal with the superlinear growth conditions. Precisely, the main result is related to the existence of nontrivial positive radial solutions.

Section snippets

The sublinear case

We first give some standard definitions and notations. For 1<m<N, let m*=Nm/(N-m) be the critical Sobolev exponent of m.

Let D1,m(RN) be the closure of C0(RN) with respect to the norm u1,mum=(RN|u|mdx)1/m. D1,m(RN) is a uniformly convex Banach space and may be written as D1,m(RN)={uLm*(RN):u(Lm(RN))N}. Moreover, Sobolev imbedding holds; in fact, there exists a positive constant c such that um*cu1,m for all uD1,m(RN).

We denote by Z the product space D1,p(RN)×D1,q(RN) ; Z*

The superlinear case

The goal of this section is to establish the existence of positive radial solutions to system (S) under superlinear growth conditions. Special choice of the nonlinearities f and g offers new tools to achieve our existence proof. Namely, the method of solvability refers to the statement of [9]. Precisely, we utilize the following theorem. Theorem (see [9, p. 56]). Let K be a cone in a Banach space X and L:KK a compact map such that L(0)=0. Assume that there exist numbers 0<ρ<θ such that

  • (i)

    UσL(U)

Acknowledgements

Our research has been generously supported by grants from ANDRU under No. CU39904.

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