Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
📌 3 Maddede Bu Makale
🎯
Amaç & Hipotez:In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type $$ \begin{cases} - M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\ u = 0, \quad x\in \partial\Omega, \end{cases} $$ where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$.
📊
Temel Bulgular & Veri:In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type $$ \begin{cases} - M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\ u = 0, \quad x\in \partial\Omega, \end{cases} $$ where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$.
💡
Sonuç & Katkı:Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Anisotropic elliptic equations · Kirchhoff type equations · Variable exponents
Abstract
In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type $$ \begin{cases} - M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\ u = 0, \quad x\in \partial\Omega, \end{cases} $$ where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$. Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
Chung, N. T. (2018). Multiple solutions for an anisotropic elliptic equation of Kirchhoff type in bounded domain. Results in Nonlinear Analysis, Cilt: 1 Sayı: 3, 116–127. https://ajindex.com/articles/1537769
G.A. Afrouzi, M. Mirzapour, Existence and multiplicity of solutions for nonlocal −!p (x)-Laplacian problem,Taiwanese J. Math., 18(1) (2014), 219-236.
G.A. Afrouzi, M. Mirzapour, V.D. Radulescu, Qualitative analysis of solutions for a class of anisotropicelliptic equations with variable exponent, Proc. Edinburgh Math. Soc., 59(3) (2016), 541-557.
G.A. Afrouzi, M. Mirzapour, V.D. Radulescu, Variational analysis of anisotropic Schrodinger equationswithout Ambrosetti-Rabinowitz-type condition, Z. Angew. Math. Phys., 69(1) (2017), 1-17.
A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical points theory and applications, J.Funct. Anal., 04 (1973), 349-381.
M. Avci, On a nonlocal problem involving the generalized anisotropic −!p (:)-Laplace operator, Annals of theUniversity of Craiova, Mathematics and Computer Science Series, 43(2) (2016), 259-272.
M. Avci, R.A. Ayazoglu (Mashiyev), B. Cekic, Solutions of an anisotropic nonlocal problem involvingvariable exponent, Adv. Nonlinear Anal., 2(3) (2013), 325-338.
A. Bensedik, On existence results for an anisotropic elliptic equation of Kirchhoff-type by a monotonicitymethod, Funkcialaj Ekvacioj, 57(3) (2014), 489-502.
M.M. Boureanu, V. R˘adulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent,Nonlinear Anal. (TMA), 75 (2012), 4471-4482.
F. Cammaroto, L. Vilasi, Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacianoperator, Nonlinear Anal. (TMA), 74 (2011), 1841-1852.
A.D. Castro, E. Montefusco, Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations,Nonlinear Anal. (TMA), 70 (2009), 4093-4105.
S.J. Chen, L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on RN, Nonlinear Anal.(TMA), 14(2013), 1477-1486.
M. Chipot, B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. (TMA),30 (7) (1997), 4619-4627.
N.T. Chung, Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities, Complex Var. Elliptic Equa., 58(12) (2013), 1637-1646.
N.T. Chung, H.Q. Toan, On a class of anisotropic elliptic equations without Ambrosetti-Rabinowitz typeconditions, Nonlinear Anal. (RWA), 16 (2014), 132-145.
F.J.S.A. Correa, G.M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bull.Aust. Math. Soc., 74 (2006), 263-277.
F.J.S.A. Correa, G.M. Figueiredo, On a p-Kirchhoff equation via Krasnoselskii’s genus, Appl. Math. Letters,22 (2009), 819-822.
G. Dai, R. Hao, Existence of solutions for a p(x)-Kirchhoff-type equation, J. Math. Anal. Appl., 359 (2009),275-284.
L. Diening, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents,Lecture Notes, vol. 2017, Springer-Verlag, Berlin, 2011.
D.S. Dumitru, Two nontrivial solutions for a class of anisotropic variable exponent problems, Taiwanese J.Math., 16(4) (2012), 1205-1219.
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
I. Fragal`a, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear equations,Ann. Inst. H. Poincar´e, Analyse Non Lin´eaire, 21 (2004), 715-734.
O. Kov´aˇcik, J. R´akosn´ık, On spaces Lp(x) and W 1;p(x), Czechoslovak Math. J., 41 (1991), 592-618.
G. Kirchhoff, Mechanik, Teubner, Leipzig, Germany, 1883.
J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Proceedings ofinternational Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro 1977.
M. Mihailescu, P. Pucci, V.D. R˘adulescu, Eigenvalue problems for anisotropic quasilinear elliptic equationswith variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698.
P. Pucci, M. Xiang, B. Zhang, Multiple solutions for nonhomogeneous Schrodinger-Kirchhoff type equationsinvolving the fractional p-Laplacian in RN, Calc. Var., 54(3), (2015), 2785-2806.
M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2002.
J. Simon, R´egularit´e de la solution d’une ´equation non lin´eaire dans RN, in: Ph. B´enilan, J. Robert (Eds.),Journ´ees d’Analyse Non Lin´eaire, in: Lecture Notes in Math., Vol. 665, Springer-Verlag, Berlin, 1978,205-227.
M. Yu, L. Wang, S. Tang, Existence of solutions for an anisotropic elliptic problem with variable exponentand singularity, Math. Methods in the Appl. Sci., 39(10) (2016), 2761-2767.