Existence, multiplicity and regularity of solutions for the fractional $p$-Laplacian equation
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We are concerned with the following elliptic equations: \begin{equation*} \begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \textmd{in} \ \ \Omega,\\ u= 0\quad \text{on}\ \ \mathbb{R}^N\backslash\Omega, \end{cases} \end{equation*} where $\lambda$ are real parameters, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\Omega\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters $\lambda$ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity $f$ has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in $L^{\infty}(\Omega)$ of any possible weak solution by applying the bootstrap argument.
Abstract
We are concerned with the following elliptic equations: \begin{equation*} \begin{cases} (-\Delta)_p^su=\lambda f(x,u) \quad \textmd{in} \ \ \Omega,\\ u= 0\quad \text{on}\ \ \mathbb{R}^N\backslash\Omega, \end{cases} \end{equation*} where $\lambda$ are real parameters, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $0<s<1<p<+\infty$, $sp<N$, and $f:\Omega\times\mathbb R \to \mathbb R$ satisfies a Carath\'eodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters $\lambda$ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity $f$ has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in $L^{\infty}(\Omega)$ of any possible weak solution by applying the bootstrap argument.
참고문헌 (36)
- R. A. Adams, "Pure and Applied Mathematics (Amsterdam), 140", pp.2003, Elsevier
- A. Ambrosetti, "Dual variational methods in critical point theory and applications", J. Functional Analysis, 14, pp.349~381 (1973)[DOI]
- J. -H. Bae, "Critical points theorems via the generalized Ekeland variational principle and its application to equations of p(x)-Laplace type in RN", Taiwanese J. Math, 23(1), pp.193~229 (2019)[DOI]
- G. Barletta, "Existence results for a Neumann problem involving the p(x)-Laplacian with discontinuous nonlinearities", Nonlinear Anal. Real World Appl, 27, pp.312~325 (2016)[DOI]
- B. Barrios, "On some critical problems for the fractional Laplacian operator", J. Dierential Equations, 252(11), pp.6133~6162 (2012)[DOI]
- J. Bertoin, "Cambridge Tracts in Mathematics, 121" (1996), Cambridge Univer-sity Press
- Z. Binlin, "Superlinear nonlocal fractional problems with innitely many solutions", Nonlinearity, 28(7), pp.2247~2264 (2015)[DOI]
- C. Bjorland, "Non-local gradient dependent operators", Adv. Math, 230(4-6), pp.1859~1894 (2012)[DOI]
- G. Bonanno, "A critical point theorem via the Ekeland variational principle", Nonlinear Anal, 75(5), pp.2992~3007 (2012)[DOI]
- G. Bonanno, "Relations between the mountain pass theorem and local minima", Adv. Nonlinear Anal, 1(3), pp.205~220 (2012)[DOI]
- G. Bonanno, "Discontinuous elliptic problems involving the p(x)-Laplacian", Math. Nachr, 284(5-6), pp.639~652 (2011)[DOI]
- G. Bonanno, "Existence and multiplicity of weak solutions for elliptic Dirichlet problems with variable exponent", J. Math. Anal. Appl, 418(2), pp.812~827 (2014)[DOI]
- G. Bonanno, "Sturm-Liouville equations involving discontinuous nonlinearities", Minimax Theory Appl, 1(1), pp.125~143 (2016)
- L. Brasco, "Stability of variational eigenvalues for the fractional p-Laplacian", Discrete Contin. Dyn. Syst, 36(4), pp.1813~1845 (2016)[DOI]
- L. Caarelli, "Non-local diusions, drifts and games, in Nonlinear partial dierential equations", Abel Symp, 7, pp.37~52 (2012), Springer
- F. Demengel, "Uni-versitext" (2012), Springer
- E. Di Nezza, "Hitchhiker's guide to the fractional Sobolev spaces", Bull. Sci. Math, 136(5), pp.521~573 (2012)[DOI]
- G. Gilboa, "Nonlocal operators with applications to image processing", Mul-tiscale Model. Simul, 7(3), pp.1005~1028 (2008)[DOI]
- A. Iannizzotto, "Existence results for fractional p-Laplacian problems via Morse theory", Adv. Calc. Var, 9(2), pp.101~125 (2016)[DOI]
- J. -M. Kim, "Multiplicity of small or large energy solutions for Kirchho-Schrodinger-Type equations involving the fractional p-Laplacian in RN", Sym-metry, 10, pp.1~21 (2018)[DOI]
- N. Laskin, "Fractional quantum mechanics and Levy path integrals", Phys. Lett. A, 268(4-6), pp.298~305 (2000)[DOI]
- N. Laskin, "Fractional Schrodinger equation", Phys. Rev. E, 66(5), pp.7~ (2002)[DOI]
- R. Lehrer, "On fractional p-Laplacian problems with weight", Dierential Integral Equations, 28(1-2), pp.15~28 (2015)
- S. Liu, "On superlinear problems without the Ambrosetti and Rabinowitz condition", Non-linear Anal, 73(3), pp.788~795 (2010)[DOI]
- S. Liu, "Innitely many solutions for a superlinear elliptic equation", Acta Math. Sinica(Chin. Ser. ), 46(4), pp.625~630 (2003)
- V. Maz'ya, "On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces", J. Funct. Anal, 195(2), pp.230~238 (2002)[DOI]
- R. Metzler, "The random walk's guide to anomalous diusion : a fractional dynamics approach", Phys. Rep, 339(1), pp.77~ (2000)[DOI]
- R. Metzler, "The restaurant at the end of the random walk : recent developments in the description of anomalous transport by fractional dynamics", J. Phys. A, 37(31), pp.R161~R208 (2004)[DOI]
- O. H. Miyagaki, "Superlinear problems without Ambrosetti and Rabinowitz growth condition", J. Dierential Equations, 245(12), pp.3628~3638 (2008)[DOI]
- R. Pei, "Existence results for asymmetric fractional p-Laplacian problem", Math. Nachr, 290(16), pp.2673~2683 (2017)[DOI]
- K. Perera, "Bifurcation and multiplicity results for critical fractional p-Laplacian problems", Math. Nachr, 289(2-3), pp.332~342 (2016)[DOI]
- R. Servadei, "Innitely many solutions for fractional Laplace equations with subcritical nonlinearity, in Recent trends in nonlinear partial dierential equations. II. Stationary problems", Contemp. Math, 595, pp.317~340 (2013)[DOI]
- R. Servadei, "Mountain pass solutions for non-local elliptic operators", J. Math. Anal. Appl, 389(2), pp.887~898 (2012)[DOI]
- Y. Wei, "Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian", Calc. Var. Partial Dierential Equations, 52(1-2), pp.95~124 (2015)[DOI]
- M. Xiang, "Existence of solutions for Kirchhotype problem involving the non-local fractional p-Laplacian", J. Math. Anal. Appl, 424(2), pp.1021~1041 (2015)[DOI]
- B. Zhang, "Multiplicity of solutions for a class of superlinear non-local fractional equations", Complex Var. Elliptic Equ, 60(5), pp.583~595 (2015)[DOI]