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Research Articles

Numerical solution of a critical Sobolev exponent problem with weight on 𝕊³

Adel Almarashia Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi ArabiaCorrespondence[email protected]

https://orcid.org/0000-0002-2542-3320

Idir Mechaia Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

https://orcid.org/0000-0003-4107-9010

Ahmed Msmalia Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

https://orcid.org/0000-0002-0186-0526

Habib Yazidib National School of Engineers of Tunis, University of Tunis, Tunis, Tunisia

https://orcid.org/0000-0002-5386-772X

Figures & data

Figure 1. Numerical solutions of (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) for $α = 1$ , $β_{k} = 1$ , $λ = 2$ , and different values of R and k: (a) R = 0.981918, k = 0.1. (b) R = 1.08245, k = 0.5. (c) R = 1.08481, k = 0.9. (d) R = 1.07375, k = 1.

Figure 1. Numerical solutions of (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) for α=1, βk=1, λ=2, and different values of R and k: (a) R = 0.981918, k = 0.1. (b) R = 1.08245, k = 0.5. (c) R = 1.08481, k = 0.9. (d) R = 1.07375, k = 1.

Figure 2. Numerical solutions of (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) for $α = 1$ , $β = 1$ , $λ = - 6$ , and different values of R and k: (a) R = 26.4421, k = 1.1. (b) R = 43.4183, k = 1.2. (c) R = 114.356, k = 1.3. (d) R = 316.708, k = 1.35.

Figure 2. Numerical solutions of (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) for α=1, β=1, λ=−6, and different values of R and k: (a) R = 26.4421, k = 1.1. (b) R = 43.4183, k = 1.2. (c) R = 114.356, k = 1.3. (d) R = 316.708, k = 1.35.

Figure 3. Numerical solutions of (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) for $λ = 6$ (a), (b) and $λ = - 6$ (c), (d), with different values of R, k, α, β: (a) R = 1.89151, k = 0.9, $α = 2$ , $β = 5$ ; (b) R = 1.38993, k = 0.9, $α = 3$ , $β = 4$ ; (c) R = 29.3202, k = 1.1, $α = 0.5$ , $β = 0.25$ ; (d) R = 8.90831, k = 1.3, $α = 2$ , $β = 0.2$ .

Figure 3. Numerical solutions of (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) for λ=6 (a), (b) and λ=−6 (c), (d), with different values of R, k, α, β: (a) R = 1.89151, k = 0.9, α=2, β=5; (b) R = 1.38993, k = 0.9, α=3, β=4; (c) R = 29.3202, k = 1.1, α=0.5, β=0.25; (d) R = 8.90831, k = 1.3, α=2, β=0.2.

Table 1. Minimum radius R for (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $α = 1$ , k = 0.9, $λ = 2$ and β.

Display Table

Table 2. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $β = 1$ , k = 0.9, $λ = 2$ and α.

Display Table

Table 3. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $α = 2$ , k = 1.3, $λ = - 6$ and β.

Display Table

Table 4. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $β = 0.1$ , k = 1.3, $λ = - 6$ and α.

Display Table

Numerical solution of a critical Sobolev exponent problem with weight on 𝕊³

Table 1. Minimum radius R for (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $α = 1$ , k = 0.9, $λ = 2$ and β.

Table 2. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $β = 1$ , k = 0.9, $λ = 2$ and α.

Table 3. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $α = 2$ , k = 1.3, $λ = - 6$ and β.

Table 4. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $β = 0.1$ , k = 1.3, $λ = - 6$ and α.

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Numerical solution of a critical Sobolev exponent problem with weight on 𝕊3

Figures & data

Table 1. Minimum radius R for (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) with α=1, k = 0.9, λ=2 and β.

Table 2. Minimum radius R (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) with β=1, k = 0.9, λ=2 and α.

Table 3. Minimum radius R (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) with α=2, k = 1.3, λ=−6 and β.

Table 4. Minimum radius R (Equation12(12) {−(ρ(r)p(r)r2u′(r))′=r2ρ2(r)u5(r)+λr2ρ3(r)u(r),forr∈(0,R),u′(0)=u(R)=0,(12) ) with β=0.1, k = 1.3, λ=−6 and α.

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Numerical solution of a critical Sobolev exponent problem with weight on 𝕊³

Table 1. Minimum radius R for (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $α = 1$ , k = 0.9, $λ = 2$ and β.

Table 2. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $β = 1$ , k = 0.9, $λ = 2$ and α.

Table 3. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $α = 2$ , k = 1.3, $λ = - 6$ and β.

Table 4. Minimum radius R (Equation12(12) ${\begin{cases} - {(ρ (r) p (r) r^{2} u^{'} (r))}^{^{'}} = r^{2} ρ^{2} (r) u^{5} (r) \\ + λ r^{2} ρ^{3} (r) u (r), for r \in (0, R), \\ u^{'} (0) = u (R) = 0, \end{cases}$ (12) ) with $β = 0.1$ , k = 1.3, $λ = - 6$ and α.