Solutions to indefinite weakly coupled cooperative elliptic systems
DOI:
https://doi.org/10.12775/TMNA.2020.052Keywords
Weakly coupled elliptic system, indefinite, cooperative, subcritical, critical, existence and multiplicity of solutionsAbstract
We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 - \kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\beta u_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 + \lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\ u_1,u_2\in D^{1,2}_0(\Omega), \end{cases} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb R^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb R$, $\mu_1,\mu_2,\lambda> 0$, $\alpha,\beta> 1$, and $\alpha + \beta = p\le 2^*:={2N}/({N-2})$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $\lambda$ sufficiently large. If $p=2^*$ and $\kappa_1,\kappa_2> 0$ we establish the existence of a ground state for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.References
P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity, Nonlinear Anal. 7 (1983), no. 9, 981–1012.
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012), no. 2, 515–551.
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calc. Var. Partial Differential Equations 52 (2015), no. 1–2, 423–467.
M. Clapp and J. Faya, Multiple solutions to a weakly coupled purely critical elliptic system in bounded domain, Discrete Contin. Dyn. Syst. 39 (2019), no. 6, 3265–3289.
F. Gazzola and B. Ruf, Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Diff. Equations 2 (1997), 555–572.
T.-C. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys. 255 (2005), no. 3, 629–653.
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math. 73 (2005), 259–287.
S. Peng, Y. Peng, Z.-Q. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations 55 (2016), no. 6, Art. 142, 30 pp.
A. Pistoia and H. Tavares, Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions, J. Fixed Point Theory Appl. 19 (2017), no. 1, 407–446.
N. Soave and H. Tavares, New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms, J. Differential Equations 261 (2016), no. 1, 505–537.
A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal. 257 (2009), no. 12, 3802–3822.
A. Szulkin, T. Weth and M. Willem, Ground state solutions for a semilinear problem with critical exponent, Differential Integral Equations 22 (2009), no. 9–10, 913–926.
M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
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