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Singular elliptic systems with asymptotically linear nonlinearities. (English) Zbl 1299.35100

The paper deals with the elliptic system \[ \begin{cases} - \Delta u_1 = \mu_1 \, f_1(u_2) \; \text{ in }\Omega ,\\ - \Delta u_2 = \mu_2 \, f_2(u_1) \; \text{ in }\Omega ,\\ u_1 = u_2 = 0 \; \text{ on }\partial \Omega, \end{cases} \] where \(\Omega \) is a smooth bounded domain and \(\mu_1,\mu_2\) are positive parameters. The nonlinearities \(f_1(t),f_2(t)\) must be continuous for \(t \in (0,+\infty)\), and may have a singularity at the origin, mild in the sense that \(\limsup_{t \to 0^{+}} t^{\gamma }\, | f_i(t)| < \infty \) for \(i = 1,2\) with a convenient exponent \(\gamma \in (0,1)\). The nonlinearities are called asymptotically linear in the sense that \(f_i(t) \sim m_i \, t\) as \(t \to +\infty \) for convenient \(m_i \in (0,+\infty ),\;i = 1,2\).
Existence of positive solutions is investigated. To give an idea, assume that \(f_1(t),f_2(t)\) are positive for every \(t\). If \(\mu_1 \, \mu_2 < \frac {\lambda_1^2}{m_1 \, m_2}\), where \(\lambda_1\) is the first eigenvalue of the Dirichlet-Laplacian in \(\Omega \), then the above problem admits at least one solution made up of a couple of positive functions \(u_1,u_2\) in the Hölder class \(C^{1,\alpha }(\overline \Omega)\) for some \(\alpha \in (0,1)\).
Related results were obtained in [C. Peng and J. Yang, Glasg. Math. J. 49, No. 2, 377–390 (2007; Zbl 1158.35031)]. For the single-equation case, instead, the author refers to [A. Ambrosetti, D. Arcoya and B. Buffoni, Differ. Integral Equ. 7, No. 3-4, 655–663 (1994; Zbl 0808.35030); A. Ambrosetti and P. Hess, J. Math. Anal. Appl. 73, 411–422 (1980; Zbl 0433.35026); D. D. Hai, Topol. Methods Nonlinear Anal. 39, No. 1, 83–92 (2012; Zbl 1387.35230)].
Existence is proved by applying the Schauder fixed-point theorem to the operator \(T\) that takes two functions \(v_1,v_2 \in C(\overline \Omega)\) onto the solutions \(u_1,u_2\) of the corresponding linearized system In order to satisfy the assumptions of the Schauder theorem (see, for instance, [S. Kesavan, Topics in functional analysis and applications. New York: John Wiley & Sons (1989; Zbl 0666.46001)]), the operator \(T\) is restricted to a suitable subset \(\mathbf {K} \subset C(\overline \Omega ) \times C(\overline \Omega)\) made up of (couples of) functions bounded by scalar multiples of the first positive eigenfunction \(\phi_1\) of the Dirichlet-Laplacian in \(\Omega \) (normalized with \(\| \phi_1 \|_{\infty }= 1\)). Then, using the comparison principle, the author shows that the image of \(\mathbf {K}\) by the operator \(T\) is included in \({\mathbf K}\). The proof of compactness of the operator \(T\) is based on a preliminary extension of the corresponding well-known property of the inverse operator of the Dirichlet-Laplacian to the case when the right-hand side \(h(x)\) belongs to \(L^1(\Omega)\) and is dominated by \(C\operatorname{dist}(x,\partial \Omega ))^{-\gamma }\) for some \(C > 0\) and for \(x\) close to \(\partial \Omega \).
In the last part of the paper, the author investigates nonexistence. More precisely, he starts from the assumption that a couple \((u_1,u_2)\) of positive solutions exists. Then, by integrating the equations previously multiplied by \(\phi_1\), and by assuming that \(f_i(t) \geq m_i \, t\) for \(t > 0\), \(i = 1,2\), the author derives the inequality \(\mu_1 \, \mu_2 \leq \frac {\lambda_1^2}{m_1 \, m_2}\) (written with \(<\) instead of \(\leq \) in the paper). Hence the reversed inequality gives a nonexistence criterion. Further nonexistence criteria for systems are given in [D.-P. Covei, Appl. Math. Lett. 25, No. 3, 610–613 (2012; Zbl 1254.35075)].

MSC:

35J15 Second-order elliptic equations
35J50 Variational methods for elliptic systems
35B06 Symmetries, invariants, etc. in context of PDEs
35B09 Positive solutions to PDEs
35B35 Stability in context of PDEs
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