Radial Positive Solutions for Problems Involving φ-Laplacian Operators with Weights
DOI:
https://doi.org/10.15407/mag20.02.153Анотація
Використовуючи теорiю потенцiалу, встановлюємо iснування та асимптотичну поведiнку радiальних розв’язкiв наступної крайової задачі:
\begin{equation*}
\left\{
\begin{aligned}
&-\dfrac{1}{A}(A\phi(\mid u' \mid) u')'=a(t)u^\sigma & \mbox{on } (0,1),\\
&A \phi(\mid u'\mid )u'(0)=0, \\
&u(1)=0,
\end{aligned}
\right.
\end{equation*}
де $\sigma>0$, $A$ є додатною диференційовною функцією на $(0,1)$, а невід’ємна функція $\phi$ є неперервно диференційовною на $[0,\infty)$ так, що для кожного $t>0$, $$k_1 \le \dfrac{(t\phi(t))'}{\phi(t)} \le k_2,$$ де $k_1>0$ і $k_2>0$. Невід'ємна нелінійність $a$ повинна задовольняти деякі відповідні припущення, пов’язані з теорією регулярних варіацій Карамати. Ми закінчуємо цю роботу розглядом застосувань.
Mathematical Subject Classification 2020: 26A12, 34A34, 34B15, 34B18, 34B27
Ключові слова:
додатнi розв’язки, асимптотична поведiнка, φ-лапласiан, клас КараматиПосилання
R.P. Agarwal, H. Lu, and D. O'Regan, Existence theorems for the one-dimensional singular $p$-Laplacian equation with sign changing nonlinearities, Appl. Math. Comput. 143 (2003), 15--38. https://doi.org/10.1016/S0096-3003(02)00342-9
I. Bachar, S. Ben Othman, and H. Mâagli, Radial solutions for the $p$-Laplacian equation, Nonlinear Anal. 70 (2009), 2198--2205. https://doi.org/10.1016/j.na.2008.02.119
V. Benci, D. Fortunato, and L. Pisani, Soliton like solutions of a Lorentz invariant equation in dimension 3, Rev. Math. Phys. 10 (1998), 315--344. https://doi.org/10.1142/S0129055X98000100
S. Ben Othman and H. Mâagli, Asymptotic behavior of positive solutions for the radial $p$-Laplacian equation, Electron. J. Differential Equations, 2012 (2012), No. 240, 1--10.
S. Ben Othman, R. Chemmam, and H. Mâagli, Asymptotic behavior of ground state radial soloutions for $p$-Laplacian problems, J. Math. 2013 (2013), 1--7. https://doi.org/10.1155/2013/409329
C. Bereanu and J. Mawhin, Boundary value problems for some nonlinear systems with singular φ-Laplacian, J. Fixed Point Theory Appl. 4 (2008), 57--75. https://doi.org/10.1007/s11784-008-0072-7
R. Chemmam, H. Mâagli, S. Masmoudi, and M. Zribi, Combined effects in nonlinear singular elliptic problems in a bounded domain, Adv. Nonlinear Anal. 4 (2012), 301--318. https://doi.org/10.1515/anona-2012-0008
A. Chinni, B. Di Bella, P. Jebelean, and R. Precup, A four-point boundary value problem with singular φ-Laplacian, J. Fixed Point Theory Appl. 2 (2019), 1--16. https://doi.org/10.1007/s11784-019-0703-1
H. Dang and S.F. Oppenheimer, Existence and uniqueness results for some nonlinear boundary value problems, J. Math. Anal. Appl. 98 (1996), 35--48. https://doi.org/10.1006/jmaa.1996.0066
A. Dhifli, R. Chemmam, and S. Masmoudi. Asymptotic behavior of ground state radial solutions for problems involving the φ-Laplacian, Positivity, 24 (2020), 957--971. https://doi.org/10.1007/s11117-019-00715-y
S. Dridi, B. Khamessi, S. Turki, and Z. Zine El Abidine, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Stud. 22 (2015), 87--103.
N. Fukagai and K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Ann. di Mat. Pura ed Appl. 186 (2007), 539--564. https://doi.org/10.1007/s10231-006-0018-x
X. He and W. Ge, Twin positive solutions for the one-dimensional $p$-Laplacian boundary value problems, Nonlinear Anal. Theory Methods Appl. 56 (2004), 975--984. https://doi.org/10.1016/j.na.2003.07.022
H. Lü, D. O'regan, and R. P. Agrawal, Existence theorems for the one dimensional singular $p$-Laplacian equations with a nonlinear boundary condition, J. Comput. Appl. Math. 182 (2005), 188--210. https://doi.org/10.1016/j.cam.2004.10.022
M. Maric, Regular Variation and Differential Equations, Lecture notes in mathematics, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/BFb0103952
R. Precup and J. Rodriguez-Lopez, Positive solutions for discontinuous problems with applications to φ-Laplacian equations , J. Fixed Point Theory Appl. 20 (2018), 1--17. https://doi.org/10.1007/s11784-018-0636-0
P. Pucci, M. Garcia-Huidobro, R. Manásevich, and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights, Ann. di Mat. Pura ed Appl. 5 (2006), 205--243. https://doi.org/10.1007/s10231-004-0143-3
W. Reichel and W. Walter, Radial solutions of equations and inequalities involving the $p$-Laplacian, J. Inequal. Appl. 1 (1997), 47--71. https://doi.org/10.1155/S1025583497000040
C. A. Santos, J. Zhou, and J. Abrantes Santos, Necessary and sufficient conditions for existence of blow-up solutions for elliptic problems in Orlicz-Sobolev spaces, Math. Nachr. 291 (2018), 160--177. https://doi.org/10.1002/mana.201600231
R. Seneta, Regular Varying Functions, Lectures in Mathematics, Springer-Verlag, Berlin, 1976. https://doi.org/10.1007/BFb0079658
