Existence results for quasilinear elliptic systems in R^{<small>N</small>}

• Main Authors: N. M. Stavrakakis, N. B. Zographopoulos
• Format: Article
• Language: English
• Published: Texas State University 1999-10-01
• Series: Electronic Journal of Differential Equations
• Subjects: p-Laplacian; nonlinear eigenvalue problems; homogeneous Sobolev spaces; maximum principle; Palais-Smale Condition.
• Online Access: http://ejde.math.txstate.edu/Volumes/1999/39/abstr.html
• ISSN: 1072-6691

We prove existence results for the quasilinear elliptic system $$ displaylines{ -Delta_{p}u = lambda a(x)|u|^{gamma-2}u +lambda b(x) |u|^{alpha -1}|v|^{Beta +1}u, cr -Delta_{q}v = lambda d(x)|v|^{delta-2}v +lambda b(x)|u|^{alpha +1}|v|^{Beta -1}v,, cr }$$ where $gamma$ and $delta$ may reach the critical Sobolev exponents, and the coefficient functions $a$, $b$, and $d$ may change sign. For the unperturbed system ($a=0$, $b=0$), we establish the existence and simplicity of a positive principal eigenvalue, under the assumption that $u(x)>0$, $v(x)>0$, and $lim_{|x| o infty} u(x) =lim_{|x| o infty} u(x)=0$.