Existence of solutions for the semilinear corner degenerate elliptic equations
Keywords
Semilinear, corner, degenerate ellipticAbstract
In this paper, we are concerned with the following elliptic equations: \begin{equation*}\label{e:JG} \begin{cases} -\Delta_{\mathbb{M}}u = \lambda f &\textmd{in } z:= (r,x,t) \in \mathbb{M}_0,\\ u= 0 &\text{on } \partial\mathbb{M}. \end{cases} \end{equation*} Here, $\lambda > 0$ and $M=[0,1)\times X\times[0,1)$ as a local model of stretched corner-manifolds, that is, the manifolds with corner singularities with dimension $N=n+2\geq3$. Here $X$ is a closed compact submanifold of dimension $n$ embedded in the unit sphere of $\mathbb{R}^{n+1}$. We study the existence of nontrivial weak solutions for the semilinear corner degenerate elliptic equations without the Ambrosetti and Rabinowitz condition via the mountain pass theorem and fountain theorem.References
C.O. Alves and S.B. Liu, On superlinear p(x)-Laplacian equations in R, Nonlinear Anal. 73 (2010), 2566–2579.
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.
H. Chen, X. Liu and Y. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations, J. Funct. Anal. 266 (2014), no. 6, 3815–3839.
H. Chen, S.Y. Tian and Y.W. Wei, Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term, J. Funct. Anal. 270 (2016), no. 4, 1602–1621.
J.V. Egorov and B.W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Oper. Theory Adv. Appl., vol. 93, Birkhäuser–Verlag, Basel, 1997.
X. Fan and X. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in RN , Nonlinear Anal. 59 (2004), 173–188.
V.K. Le, On a sub-supersolution method for variational inequalities with Leray–Lions operators in variable exponent spaces, Nonlinear Anal. 71 (2009), 3305–3321.
X. Lin, X.H. Tang, Existence of infinitely many solutions for p-Laplacian equations in R, Nonlinear Anal. 92 (2013), 72–81.
S.B. Liu, On ground states of superlinear p-Laplacian equations in R, J. Math. Anal. Appl. 361 (2010), 48–58.
S.B. Liu and S.J. Li Infinitely many solutions for a superlinear elliptic equation, Acta Math. Sinica (Chin. Ser.) 46 (2003), 625–630. (in Chinese)
R. Mazzeo, Elliptic theory of differential edge operators, Comm. Partial Differential Equations 16 (1991), 1616–1664.
R.B. Melrose and P. Piazza, Analytic K-theory on manifolds with corners, Adv. Math. 23 (1974),729–754.
O.H. Miyagaki and M.A.S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), 3628–3638.
M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0