Sign-changing solutions for the boundary value problem involving the fractional $p$-Laplacian
Keywords
Fractional $p$-Laplacian, sign-changing solutions, topology degree, deformation lemmaAbstract
In the paper, we consider the following boundary value problem involving the fractional $p$-Laplacian: \begin{equation} \tag{$\mathcal{P}$} \begin{cases} (-\triangle)_p^su(x)=f(x,u) &\text{in } \Omega,\\ u(x)=0 &\text{in } \mathbb{R}^N\setminus\Omega. \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ with $N\geq 1$, $(-\Delta)_p^{s}$ is the fractional $p$-Laplacian with $s\in (0,1)$, $p\in(1,{N}/{s})$, $f(x, u)\colon \Omega\times\mathbb{R}\rightarrow\mathbb{R}$. Under the improved subcritical polynomial growth condition and other conditions, the existences of a least-energy sign-changing solution for the problem $(\mathcal{P})$ has been established.References
C.Z. Bai, Existence results for non-local operators of elliptic type, Nonlinear Anal. 83 (2013), 82–90.
B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations 252 (2012), 6133–6162.
T. Bartsch, Z.L. Liu and T. Weth, Sign changing solutions of superlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), 25–42.
T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 259–281.
H. Berestycki and P.L. Lions, Nonlinear scalar field equations II, Existence of infinitely many solutions, Arch Rational Mech. Anal. 82 (1983), 347–375.
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.
L.A. Caffarelli, J.M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 (2010), 1111–1144.
A. Capella, Solutions of a pure critical exponent problem involving the half-Laplacian in annularshaped domains, Commun. Pure Appl. Anal. 10 (2011), 1645–1662.
X. Chang, Z. Nie and Z.Q. Wang, Sign-changing solutions of fractional p-Laplacian problems, Adv. Nonlinear Stud. 19 (2019), 29–58.
X. Chang and Z.Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity 26 (2013), 479–494.
X. Chang and Z.Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations 256 (2014), 2965–2992.
M. Chermisi and E. Valdinoci, A symmetry result for a general class of divergence form PDEs in fibered media, Nonlinear Anal. 73 (2010), 4695–703.
M. Chermisi and E. Valdinoci, Fibered nonlinearities for p(x)-Laplace equations, Adv. Calc. Var. 2 (2009), 185–205.
W. Craig and D.P. Nicholls, Travelling two and three dimensional capillary gravity water waves, SIAM J. Math. Anal. 32 (2000), 323–359.
Y.B. Deng, Yinbin and W. Shuai, Sign-changing solutions for non-local elliptic equations involving the fractional Laplacian, Adv. Differential Equations 23 (2018), 109–134.
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
M.F. Furtado, L.A. Maia and E.S. Medeiros, Positive and nodal solutions for a nonlinear Schrödinger equation with indefinite potential, Adv. Nonlinear Stud. 8 (2008), 353–373.
C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal. 17 (1986), 803–835.
E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math. 217 (2008), 1301–1312.
E.S. Noussair and J.C. Wei, On the effect of domain geometry on the existence of nodal solutions in singular perturbations problems, Indiana Univ. Math. J. 46 (1997), 1255–1271.
C. Qiu, Y.S. Huang and Y.Y. Zhou, Optimization problems involving the fractional laplacian, Electron. J. Differential Equations 98 (2016), 1–15.
W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations 259 (2015), 1256–1274.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112.
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), 842–1864.
J.F. Toland, The Peierls–Nabarro and Benjamin–Ono equations, J. Funct. Anal. 145 (1997), 136–150.
L. Wang, B.L. Zhang and K. Cheng, Ground state sign-changing solutions for the Schröinger–Kirchhoff equation in R3 , J. Math. Anal. Appl. 466 (2018), 1545–1569.
M. Willem, Progress in nonlinear differential equations and their applications, vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996.
L.J. Xi, Y.S. Huang and Y.Y. Zhou, The multiplicity of nontrivial solutions for hemivariational inequalities involving nonlocal elliptic operators, Nonlinear Anal. 21 (2015), 87–98.
L.J. Xi and Y.Y. Zhou, Infinitely many solutions for hemivariational inequalities involving the fractional Laplacian, J. Inequal. Appl. 302 (2019), 1–23.
M.Q. Xiang, B.l. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), 1021–1041.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
