Ground states of semilinear elliptic equations: a geometric approach

Rodrigo Bamón; Isabel Flores; Manuel del Pino

doi:10.1016/s0294-1449(00)00126-8

JournalsaihpcVol. 17, No. 5pp. 551–581

Ground states of semilinear elliptic equations: a geometric approach

Rodrigo Bamón
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653 Correo 1, Santiago, Chile
Isabel Flores
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653 Correo 1, Santiago, Chile
Manuel del Pino
Departamento de Ingenierı́a Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

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Abstract

We consider the problem

Δ u + u^{p} + u^{q} = 0, in R^{N} 0 < u (x) \to 0 as ∣ x ∣ \to + \infty

where $1 < p < (N + 2) / (N - 2) < q$ . We prove that if $q$ is fixed and we let $p$ approach $(N + 2) / (N - 2)$ from below, then this problem has a large number of radial solutions. A similar fact takes place if we fix $p > N / (N - 2)$ and then let $q$ approach $(N + 2) / (N - 2)$ . If we fix $q$ and then let $p$ be close enough to $N / (N - 2)$ then no solutions exist.

Résumé

On considère le problème de trouver des solutions de l’equation elliptique

Δ u + u^{p} + u^{q} = 0, dans R^{N}

avec

0 < u (x) \to 0 lorsque ∣ x ∣ \to + \infty

où $1 < p < (N + 2) / (N - 2) < q$ . Si l’on fixe $q$ et $p$ augmente et tend vers $(N + 2) / (N - 2)$ alors il’y a un grand nombre des solutions radials. On peut obtenir un résultat analogue si l’on fixe $p > N / (N - 2)$ et $q$ s’approche de $(N + 2) / (N - 2)$ . En plus, si l’on fixe $q$ et l’on prend $p$ assez proche de $N / (N - 2)$ alors il n’existe pas de solution.

Cite this article

Rodrigo Bamón, Isabel Flores, Manuel del Pino, Ground states of semilinear elliptic equations: a geometric approach. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 5, pp. 551–581

DOI 10.1016/S0294-1449(00)00126-8