By Alfonso Castro

The authors offer a whole class of the radial ideas to a category of response diffusion equations bobbing up within the research of thermal buildings corresponding to plasmas with thermal equilibrium or no flux on the boundary. specifically, their examine comprises swiftly growing to be nonlinearities, that's, these the place an exponent exceeds the severe exponent. They describe the corresponding bifurcation diagrams and make certain lifestyles and specialty of flooring states, which play a vital position in characterizing these diagrams. in addition they supply info at the stability-unstability of the radial regular states

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Let d ∈ (D − δ, D). Since E (t) < 0 we obtain, for T ≤ t ≤ tˆ(d), that v (t)2 ≥ F (v(tˆ)) − F (v(t)) > F ( ). 2 Hence, v (t) < − −2F ( ). It follows from this that v(T ) − v(tˆ) < tˆ − T = −v (ξ) −2F ( ) = 2−q 2 (2 − p−q ) 12 . Hence, tˆ < T + 2−q 2 (2 − p−q ) 21 . This actually shows that tˆ(d) → ξ(D) as d → D. We can now use Lemma 18 to obtain that 0 < t1 (d) ≤ c1 for d ∈ (D − δ, D). It follows from Lemma 20 that 0 < tj (d) < cj for d ∈ (D − δ, D), and j = 1, 2, . . This ﬁnished the proof of 3.

26) √ for r ≥ r0 and r0 suﬃciently large. 7)). This proves that A = lim −tz (t) ∈ [0, 2/n). 27) t→∞ Let γ ∈ (A, 2/n). By the deﬁnition of A, there exists t4 > t1 such that −tz (t) ≤ γ for t ≥ t4 . Hence z(t) ≥ z(t4 )−ln(t/t4 )γ . Without loss of generality we may assume that 2emz(t) < enz(t) for t ≥ t4 . 28) 1−nγ nz(t4 ) tnγ e ds 4 s ≤ −t4 z (t4 ) − (1/2) t4 ) tnγ enz(t4 ) (t2−nγ − t2−nγ 4 = −t4 z (t4 ) − 4 2(2 − nγ) → −∞ as t → +∞, which contradicts that z is monotonically decreasing. 2) cannot have monotonically decreasing solutions, which proves the Theorem.

If not (v (s))2 ≥ v q+1 (s)((2/(q + 1)) − (2/((p + 1))v p−q (s)) ≥ cv q+1 (s) for s > s1 , with c = (1 − q)/(2(q+1)). Thus −v ≥ cv (q+1)/2 . Integrating we have v (1−q)/2 (t) ≤ v (1−q)/2 (s1 )− c(t−s1 ) < 0 which contradicts that v(·) > 0 on (0, ∞). 104) (v (s))2 + 2F (v(s)) < 0, for all s > sˆ. 106) t 2q − 2 )) dr > 0, q +1 s which contradicts that v < 0 on (0, ∞). This proves Lemma 21. 1. 2) Δu(x) + f (u(x)) = 0. For 0 ≤ q < p ≤ 1, the existence of compactly supported ground states was proven in H.