Semilinear parabolic equation in RN associated with critical Sobolev exponent

Bifurcation of Positive Solutions for a Semilinear Equation with Critical Sobolev Exponent

In this note we consider bifurcation of positive solutions to the semilinear elliptic boundary-value problem with critical Sobolev exponent −∆u = λu− αu + u −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 [2] showed that a lower order (non-negative) perturbation can contribute t...

Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent

where λ > 0 is a parameter, κ ∈ R is a constant, p = (N + 2)/(N − 2) is the critical Sobolev exponent, and f(x) is a non-homogeneous perturbation satisfying f ∈ H−1(Ω) and f ≥ 0, f ≡ 0 in Ω. Let κ1 be the first eigenvalue of −Δ with zero Dirichlet condition on Ω. Since (1.1)λ has no positive solution if κ ≤ −κ1 (see Remark 1 below), we will consider the case κ > −κ1. Let us recall the results f...

Fast Diffusion Equation with Critical Sobolev Exponent in a Ball

Ja n 20 04 Schrödinger equation with critical Sobolev exponent

Critical Exponents for a Semilinear Parabolic Equation with Variable Reaction

In this paper we study the blow-up phenomenon for nonnegative solutions to the following parabolic problem: ut(x, t) = ∆u(x, t) + (u(x, t)) , in Ω× (0, T ), where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and o...