Infinitely many solutions for resonant cooperative elliptic systems with sublinear or superlinear terms. (English) Zbl 1288.35234

Calc. Var. Partial Differ. Equ. 49, No. 1-2, 271-286 (2014); erratum ibid. 49, No. 1-2, 287-290 (2014).
The authors of this interesting paper investigate a class of resonant cooperative elliptic systems having the form \[ -\Delta u=\lambda u+\delta v+f(x,u,v) , \quad -\Delta v=\delta u+\gamma v+g(x,u,v) \text{ in }\Omega, \] and \(u=v=0\) in \(\partial\Omega\) with sublinear and superlinear terms, where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) (\(N\geq 3\)), and \(\lambda,\delta,\gamma\in\mathbb{R}\). It is assumed that there exists a function \(F\in C^1(\overline{\Omega }\times\mathbb{R}^2,\mathbb{R})\) such that \(\nabla F=(f,g)\), which is the cooperative case. If the condition \(\sigma (A^{\ast })\cap \sigma (-\Delta )\neq \emptyset\) holds, then the elliptic system is resonant. \(A^{\ast }\) is the matrix constructed by the coefficients \(\lambda \), \(\delta \) (first row), and \(\delta \), \(\gamma \) (second row) at the linear terms of the above stated elliptic equations, \(\sigma (A^{\ast })\) is the spectrum of \(A^{\ast }\), \[ \sigma (A^{\ast })=\{\xi ,\zeta \}= \biggl\{ \frac{\lambda +\gamma }{2}\pm \sqrt{\biggl(\frac{\lambda - \gamma }{2}\biggr)^2+\delta^2} \;\biggr\}, \] \(\sigma (-\Delta )=\{\lambda_k:k=1,2,\dots; 0<\lambda_1<\lambda_2<\dots \}\), \(\lambda_i\) – eigenvalues of the Laplacian on \(\Omega \) with zero boundary condition. The subquadratic case is considered provided that three conditions hold: (a) \(F(x,U)\geq 0\), \((x,U)\in\Omega\times\mathbb{R}^2\), and \((\nabla F,U)\leq\mu F\), \(x\in\Omega \), \(|U|\geq R_1\) (\(\mu\in [1;2]\), \(R_1>0\)); (b) \(\lim_{|U|\to 0}\frac{F}{|U|^2}=\infty \) uniformly for \(x\in\Omega \), \(F(x,U)\leq c_2|U|\), \(x\in\Omega \), \(|U|\leq R_2\) (\(c_2,U>0\)); (c) \(\liminf_{|U|\to\infty }\frac{F}{|U|}\geq d>0\) uniformly for \(x\in\Omega \).
The main statement here is that under the above stated conditions (a)–(c) and if \(F(x,U)\) is even in \(U\), then the system under consideration possesses infinitely many nontrivial solutions.
The second result is stated under the following requirements:
\(|\nabla F|\leq a_1(1+|U|^{\nu -1})\), \((x,U)\in \Omega\times\mathbb{R}^2\), \(2<\nu < 2^{\ast }=2N(N-2)^{-1}\);
\(\lim_{|U|\to\infty }\frac{F}{|U|^2}=\infty\) uniformly for \(x\in\Omega \), \(F\geq 0\) for all \((x,U)\in\Omega \times\mathbb{R}^2\);
\(\liminf_{|U|\to\infty } \frac{(\nabla F,U)-2F}{|U|^{\nu }}\geq b>0\) uniformly for \(x\in\Omega \).
Here the authors also prove existence of infinitely many nontrivial solutions under the conditions (i)–(iii), and \(F\) is even in \(U\) as well.


35J50 Variational methods for elliptic systems
35P05 General topics in linear spectral theory for PDEs
35J47 Second-order elliptic systems
35B34 Resonance in context of PDEs
Full Text: DOI


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