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Continuum of Solutions for an Elliptic Problem with Critical Growth in the Gradient
| Content Provider | Semantic Scholar |
|---|---|
| Author | Coster, Colette De Jeanjean, Louis |
| Copyright Year | 2014 |
| Abstract | We consider the boundary value problem (Pλ) u ∈ H 1 0 (Ω) ∩ L∞(Ω) : −∆u = λc(x)u+ μ(x)|∇u| + h(x), where Ω ⊂ R , N ≥ 3 is a bounded domain with smooth boundary. It is assumed that c 0, c, h belong to L(Ω) for some p > N/2 and that μ ∈ L∞(Ω). We explicit a condition which guarantees the existence of a unique solution of (Pλ) when λ < 0 and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of (P0). It crosses the axis λ = 0 if (P0) has a solution, otherwise if bifurcates from infinity at the left of the axis λ = 0. Assuming that (P0) has a solution and strenghtening our assumptions to μ(x) ≥ μ1 > 0 and h 0, we show that the continuum bifurcates from infinity on the right of the axis λ = 0 and this implies, in particular, the existence of two solutions for any λ > 0 sufficiently small. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://lmb.univ-fcomte.fr/IMG/pdf/jeanjean-jfa.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/1304.3066.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Apache Axis Apache Continuum Assumed Axis vertebra Gradient Solutions Triune continuum paradigm |
| Content Type | Text |
| Resource Type | Article |
