Nontrivial solution for a semilinear elliptic equation in unbounded domain with critical Sobolev exponent

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...

Global Solution Branch and Morse Index Estimates of a Semilinear Elliptic Equation with Super-critical Exponent

We consider the nonlinear eigenvalue problem (0.1) { −Δu = up + λu in B, u > 0 in B, u = 0 on ∂B, where B denotes the unit ball in RN , N ≥ 3, λ > 0 and p > (N + 2)/(N − 2). According to classical bifurcation theory, the point (μ1, 0) is a bifurcation point from which emanates an unbounded branch C of solutions (λ, u) of (0.1), where μ1 is the principal eigenvalue of −Δ in B with Dirichlet boun...

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...

Existence of solutions for elliptic systems with critical Sobolev exponent ∗

We establish conditions for existence and for nonexistence of nontrivial solutions to an elliptic system of partial differential equations. This system is of gradient type and has a nonlinearity with critical growth.

Fast Diffusion Equation with Critical Sobolev Exponent in a Ball