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Elliptic equations with critical Sobolev exponents in dimension 3

Olivier Druet (2002)

Annales de l'I.H.P. Analyse non linéaire

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

We consider the Yamabe type family of problems $(P_{ε}) : - Δ u_{ε} = u_{ε}^{(n + 2) / (n - 2)}$ , $u_{ε} > 0$ in $A_{ε}$ , $u_{ε} = 0$ on $\partial A_{ε}$ , where $A_{ε}$ is an annulus-shaped domain of $ℝ^{n}$ , $n \geq 3$ , which becomes thinner as $ε \to 0$ . We show that for every solution $u_{ε}$ , the energy $\int_{A_{ε}} | \nabla u_{|}^{2}$ as well as the Morse index tend to infinity as $ε \to 0$ . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on $ℝ^{n}$ , a half-space or an infinite strip. Our argument also involves a Liouville type theorem...

Displaying 1 – 10 of 10

Elliptic equations with critical Sobolev exponents in dimension 3

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Esistenza e molteplicità di soluzioni per equazioni ellittiche nonlineari

Existence and nonexistence of global solutions of degenerate and singular parabolic systems.

Existence and nonexistence of global solutions of the quasilinear parabolic equations with inhomogeneous terms.

Existence and non-existence results for a nonlinear heat equation.

Existence of infinitely many solutions for elliptic boundary-value problems with nonsymmetrical critical nonlinearity.

Existence of least energy solutions to coupled elliptic systems with critical nonlinearities.

Existence of solutions for elliptic systems with critical Sobolev exponent.

Extremal values of half-eigenvalues for $p$ -Laplacian with weights in $L^{1}$ balls.

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