Let be a bounded domain with smooth boundary. We show the existence of a positive real number such that for each and each real number the equation in in subject to the homogeneous Dirichlet boundary condition possesses a nonnegative solution . Next, we analyze the asymptotic behavior of as and we show that it converges uniformly to the distance function to the boundary of the domain.
Keywords: Weak solutionviscosity solution, nonlinear elliptic equations, asymptotic behavior, distance function to the boundary
@article{COCV_2018__24_2_569_0, author = {Mih\u{a}ilescu, Mihai and Stancu\ensuremath{-}Dumitru, Denisa and Varga, Csaba}, title = {The convergence of nonnegative solutions for the family of problems \ensuremath{-} {\ensuremath{\Delta}\protect\textsubscript{p}u} = \ensuremath{\lambda}e\protect\textsuperscript{u} as p {\textrightarrow}\ensuremath{\infty}}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {569--578}, publisher = {EDP-Sciences}, volume = {24}, number = {2}, year = {2018}, doi = {10.1051/cocv/2017048}, zbl = {1404.35311}, mrnumber = {3816405}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017048/} }
TY - JOUR AU - Mihăilescu, Mihai AU - Stancu−Dumitru, Denisa AU - Varga, Csaba TI - The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞ JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 569 EP - 578 VL - 24 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017048/ DO - 10.1051/cocv/2017048 LA - en ID - COCV_2018__24_2_569_0 ER -
%0 Journal Article %A Mihăilescu, Mihai %A Stancu−Dumitru, Denisa %A Varga, Csaba %T The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞ %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 569-578 %V 24 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017048/ %R 10.1051/cocv/2017048 %G en %F COCV_2018__24_2_569_0
Mihăilescu, Mihai; Stancu−Dumitru, Denisa; Varga, Csaba. The convergence of nonnegative solutions for the family of problems − Δpu = λeu as p →∞. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 2, pp. 569-578. doi : 10.1051/cocv/2017048. http://archive.numdam.org/articles/10.1051/cocv/2017048/
[1] Concentration phenomena for Liouville’s equation in dimension four. J. Eur. Math. Soc. (JEMS) 8 (2006) 171–180. | MR | Zbl
and ,[2] On an elliptic equation with exponential growth. Rend. Sem. Mat. Univ. Padova 96 (1996) 143–175. | Numdam | MR | Zbl
and ,[3] Limits as of and related extremal problems. Rend. Sem. Mat. Univ. Politec. Torino, special issue (1991) 15–68. | MR
, and ,[4] On a family of inhomogeneous torsional creep problems, Proc. Am. Math. Soc. 145 (2017) 4397–4409. | DOI | MR | Zbl
and ,[5] The limiting behavior of solutions to inhomogeneous eigenvalue problems in Orlicz−Sobolev spaces. Adv. Nonl. Stud. 14 (2014) 977–990. | DOI | MR | Zbl
, and ,[6] Uniform esitmates and blow-up behavior for solutions of Δu = v(x)eu in tow dimensions. Commun. Partial Differ. Equ. 16 (1992) 1223–1253. | DOI | MR | Zbl
and ,[7] Limits as p →∞ of p-Laplacian problems with a superdiffusive power-type nonlinearity: Positive and sign-changing solutions. J. Math. Anal. Appl. 372 (2010) 629–644. | DOI | MR | Zbl
and ,[8] Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), 615–22. | DOI | MR | Zbl
and ,[9] On the nonlinear equation Δu + exp u = 0 and vt = Δu + exp u. Bull. Amer. Math. Soc. 75 (1969), 132–5. | MR | Zbl
,[10] Positive solutions of quasilinear elliptic equations with critical Orlicz−Sobolev nonlinearity on RN. Funkcial.Ekvac. 49 (2006) 235–267. | DOI | MR | Zbl
, and ,[11] On a Emden-Fowler type equation. Nonlinear Anal. T.M.A. 18 (1992) 1085–1097. | DOI | MR | Zbl
and[12] Quasilinear problems with exponential growth in the reaction term. Nonlinear Analysis T.M.A. 22 (1994) 481–498. | DOI | MR | Zbl
, and ,[13] On the eigenvalues of the p-Laplacian with varying p. Proc. Amer. Math. Soc. 125 (1997) 3347–3354. | DOI | MR | Zbl
,[14] On fully nonlinear PDE’s derived from variational problems of Lp norms. SIAM J. Math. Anal. 33 (2001) 545–569. | DOI | MR | Zbl
and ,[15] Uniqueness of Lipschitz Extensions: Minimizing the Sup Norm of the Gradient. Arch. Rational Mech. Anal. 123 (1993) 51–74. | DOI | MR | Zbl
,[16] Minimization problems for Lipschitz functions via viscosity solutions. Thesis University of Jyvaskyla (1996) 1–39. | MR
,[17] On the higher eigenvalues for the ∞-eigenvalue problem. Calc. Var. Partial Differ. Equ. 23 (2005) 169–192. | DOI | MR | Zbl
and ,[18] The ∞-eigenvalue problem. Arch. Rational Mech. Anal. 148 (1999) 89–105. | DOI | MR | Zbl
, and ,[19] On a family of torsional creep problems. J. Reine Angew. Math. 410 (1990) 1–22. | MR | Zbl
,[20] On the equation div(|∇u|p−2∇u) + λ|u|p−2u = 0. Proc. Amer. Math. Soc. 109 (1990) 157–164. | MR | Zbl
,[21] On non-linear Rayleigh quotients. Potential Anal. 2 (1993) 199–218. | DOI | MR | Zbl
,[22] Some applications of the maximum principle in the problem of torsional creep. SIAM J. Appl. Math. 33 (1977) 446–455. | DOI | MR | Zbl
and ,[23] The limit as p(x) →∞ of solutions to the inhomogeneous Dirichlet problem of p(x)-Laplacian. Nonl. Anal. T.M.A. 73 (2010) 2027–2035. | DOI | MR | Zbl
and ,Cited by Sources: