Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth
We study the asymptotic behavior of radially symmetric solutions to the subcritical semilinear elliptic problem $$\displaylines{ -\Delta u = u^{\frac{N+2}{N-2}}/[\log(e+u)]^{\alpha}\quad \text{in } \Omega=B_R(0)\subset\mathbb{R}^N,\cr u>0,\quad \text{in } \Omega,\cr u=0,\quad \text{on }...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2020-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/114/abstr.html |
| _version_ | 1818617173759754240 |
|---|---|
| author | Rosa Pardo Arturo Sanjuan |
| author_facet | Rosa Pardo Arturo Sanjuan |
| author_sort | Rosa Pardo |
| collection | DOAJ |
| description | We study the asymptotic behavior of radially symmetric solutions
to the subcritical semilinear elliptic problem
$$\displaylines{
-\Delta u = u^{\frac{N+2}{N-2}}/[\log(e+u)]^{\alpha}\quad \text{in }
\Omega=B_R(0)\subset\mathbb{R}^N,\cr
u>0,\quad \text{in } \Omega,\cr
u=0,\quad \text{on } \partial \Omega,
}$$
as $\alpha\to 0^+$.
Using asymptotic estimates, we prove that there exists an explicitly defined constant
L(N,R)>0, only depending on N and R, such that
$$
\limsup_{\alpha\to0^+} \frac{\alpha u_\alpha (0)^2}
{[\log(e+u_\alpha (0))]^{1+\frac{\alpha(N+2)}{2}}}
\leq L(N,R)
\le 2^*\liminf_{\alpha\to0^+}\frac{\alpha u_\alpha (0)^2}
{[\log(e+u_\alpha (0))]^{\frac{\alpha(N-4)}2}}.
$$ |
| first_indexed | 2024-12-16T17:01:29Z |
| format | Article |
| id | doaj.art-ca578ee0eb6542ba9bf77de497193251 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-16T17:01:29Z |
| publishDate | 2020-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-ca578ee0eb6542ba9bf77de4971932512022-12-21T22:23:44ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912020-11-012020114,117Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growthRosa Pardo0Arturo Sanjuan1 Univ. Complutense de Madrid, Madrid, Spain Univ. Distrital Francisco Jose de Caldas, Bogota, Colombia We study the asymptotic behavior of radially symmetric solutions to the subcritical semilinear elliptic problem $$\displaylines{ -\Delta u = u^{\frac{N+2}{N-2}}/[\log(e+u)]^{\alpha}\quad \text{in } \Omega=B_R(0)\subset\mathbb{R}^N,\cr u>0,\quad \text{in } \Omega,\cr u=0,\quad \text{on } \partial \Omega, }$$ as $\alpha\to 0^+$. Using asymptotic estimates, we prove that there exists an explicitly defined constant L(N,R)>0, only depending on N and R, such that $$ \limsup_{\alpha\to0^+} \frac{\alpha u_\alpha (0)^2} {[\log(e+u_\alpha (0))]^{1+\frac{\alpha(N+2)}{2}}} \leq L(N,R) \le 2^*\liminf_{\alpha\to0^+}\frac{\alpha u_\alpha (0)^2} {[\log(e+u_\alpha (0))]^{\frac{\alpha(N-4)}2}}. $$http://ejde.math.txstate.edu/Volumes/2020/114/abstr.htmla priori boundspositive solutionssemilinear elliptic equationsdirichlet boundary conditionsgrowth estimatessubcritical nonlinearites |
| spellingShingle | Rosa Pardo Arturo Sanjuan Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth Electronic Journal of Differential Equations a priori bounds positive solutions semilinear elliptic equations dirichlet boundary conditions growth estimates subcritical nonlinearites |
| title | Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth |
| title_full | Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth |
| title_fullStr | Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth |
| title_full_unstemmed | Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth |
| title_short | Asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth |
| title_sort | asymptotic behavior of positive radial solutions to elliptic equations approaching critical growth |
| topic | a priori bounds positive solutions semilinear elliptic equations dirichlet boundary conditions growth estimates subcritical nonlinearites |
| url | http://ejde.math.txstate.edu/Volumes/2020/114/abstr.html |
| work_keys_str_mv | AT rosapardo asymptoticbehaviorofpositiveradialsolutionstoellipticequationsapproachingcriticalgrowth AT arturosanjuan asymptoticbehaviorofpositiveradialsolutionstoellipticequationsapproachingcriticalgrowth |