## A variational approach to noncooperative elliptic systems.(English)Zbl 0852.35039

The paper deals with the existence of non-trivial solutions for elliptic systems of the form $- \Delta u_1= \lambda u_1- \delta u_2+ f_1(x, u_1, u_2),\;- \Delta u_2= \delta u_1+ \gamma u_2- f_2(x, u_1, u_2)\text{ in }\Omega,\tag{1}$
$u_{1|_{\partial \Omega}}= 0,\quad u_{2|_{\partial \Omega}}= 0,\tag{2}$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$ and the numbers $$\lambda$$, $$\gamma$$, $$\delta$$ are real parameters, $$\gamma\leq \lambda$$, $$\delta> 0$$. The functions $$f_1, f_2: \Omega\times \mathbb{R}^2\to \mathbb{R}$$ are assumed to be Carathéodory functions, $$f_1(x, 0, 0)= 0$$, $$f_2(x, 0, 0)= 0$$, $$|f_1(x, u)|+ |f_2(x, u)|\leq c|u|^{p- 1}+ d$$ $$\forall u= (u_1, u_2)\in \mathbb{R}^2$$, a.e. $$x\in \Omega$$, for some constants $$c, d> 0$$ and $$2\leq p< 2N/(N- 2)$$ if $$N\geq 3$$ or $$2\leq p< +\infty$$ if $$N= 1, 2$$. Moreover, assume that there exists a function $${\mathcal F}: \Omega\times \mathbb{R}^2\to \mathbb{R}$$ of class $$C^1$$ in the variables $$u= (u_1, u_2)$$ such that $$(f_1, f_2)= \nabla {\mathcal F}$$ with respect to $$(u_1, u_2)$$.

### MSC:

 35J50 Variational methods for elliptic systems 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

noncooperative elliptic systems; existence
Full Text:

### References:

 [1] Rothe, F., Global existence of branches of stationary solutions for a system of reaction-diffusion equations from biology, Nonlinear Analysis, 5, 487-498 (1981) · Zbl 0471.35031 [2] Lazer, A. C.; McKenna, P. J., On steady-state solutions of a system of reaction-diffusion equations from biology, Nonlinear Analysis, 6, 523-530 (1982) · Zbl 0488.35039 [3] De, Figueiredo D. G.; Mitidieri, E., A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. math. Analysis, 17, 836-849 (1986) · Zbl 0608.35022 [4] Silva, E. A., Multiple solutions for a semilinear elliptic system, CMS Tech. Summ. Rep. No. 93-7 (1992) [5] Costa, D. G.; Magalhaes, C. A., A variational approach to subquadratic perturbations of elliptic systems, J. diff. Eqns, 111, 103-122 (1994) · Zbl 0803.35052 [6] De, Figueiredo D. G.; Felmer, P. L., On superquadratic elliptic systems (1992), (preprint) · Zbl 0820.35042 [7] Costa, D. G.; Magalhaes, C. A., Un problème elliptique non-quadratique à l’infini, C. r. hebd. Séanc. Acad. Sci Paris, 315, 1059-1062 (1992), Serie I · Zbl 0815.35022 [8] Costa, D. G.; Magalhaes, C. A., Variational elliptic problems which are nonquadratic at infinity, Nonlinear Analysis, 23, 1401-1412 (1994) · Zbl 0820.35059 [9] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, (CBMS Regional Conf. Ser. Math., Vol. 65 (1986), American Mathematical Society: American Mathematical Society Berlin) · Zbl 0152.10003 [10] Benci, V.; Rabinowitz, P. H., Critical point theorems for indefinite functionals, Invent. Math., 52, 241-273 (1979) · Zbl 0465.49006 [11] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. funct. Analysis, 14, 349-381 (1973) · Zbl 0273.49063 [12] Landesman, E. M.; Lazer, A. C., Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19, 609-623 (1970) · Zbl 0193.39203 [13] Costa D.G. & Magalhaes C.A., A unified approach to a class of strongly indefinite functionals (in preparation).; Costa D.G. & Magalhaes C.A., A unified approach to a class of strongly indefinite functionals (in preparation). · Zbl 0890.47038 [14] Cerami, G., Un criterio de esistenza per i punti critici su varietà ilimitate, Rc. Ist. Lomb. Sci. Lett., 112, 332-336 (1978) · Zbl 0436.58006 [15] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Analysis, 7, 981-1012 (1983) · Zbl 0522.58012 [16] Rudin, W., (Real and Complex Analysis (1974), McGraw-Hill: McGraw-Hill Providence, RI) · Zbl 0278.26001 [17] Goncalves, J. V.; Miyagaki, O. H., Existence of nontrivial solutions for semilinear elliptic equations at resonance, Houston J. Math., 16, 583-595 (1990) · Zbl 0731.35038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.