## 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 HĂ¶lder 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

### Keywords:

elliptic system; nonexistence; Schauder fixed-point theorem