Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent
Texas State University-San Marcos, Department of Mathematics
In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent -Δu = λu - αup + u2* -1, u > 0, in Ω, u = 0, on ∂Ω. where Ω ⊂ ℝn, n ≥ 3 is a bounded C2-domain λ > λ1, 1 < p < 2* - 1 = n+2/n-2 and α > 0 is a bifurcation parameter. Brezis and Nirenberg  showed that a lower order (non-negative) perturbation can contribute to regain the compactness and whence yields existence of solutions. We study the equation with an indefinite perturbation and prove a bifurcation result of two solutions for this equation.
Critical Sobolev exponent, Positive solutions, Bifurcation
Cheng, Y. (2006). Bifurcation of positive solutions for a semilinear equation with critical Sobolev exponent. <i>Electronic Journal of Differential Equations, 2006</i>(135), pp. 1-8.