We consider a double phase problem with a reaction consisting of a parametric critical term plus a Caratheodory perturbation which is only locally restricted near zero. We show that for all big values of the parameter the problem has at least three nontrivial smooth solutions all with sign information (two of constant sign and the third nodal). Also we establish the asymptotic behavior of the solutions as . Finally if we improve the regularity of the perturbation term, we show that there is a second nodal solution for a total of four nontrivial smooth solutions all with sign information.
A.Bahrouni, V.D.Radulescu and D.Repovs, Double phase transonic flow problems with variable growth: nonliner patterns and stationary waves, Nonlinearity, To appear.
2.
P.Baroni, M.Colombo and G.Mingione, Regularity for general functionals with double phase, Calc. Var. Partial Differential Equations57 (2018), 62. doi:10.1007/s00526-018-1332-z.
3.
V.Benci, P.D’Avenia, D.Fortunato and L.Pisani, Solitons in several space dimensions: Derrick’s problem and infinitely many solutions, Arch. Ration. Mech. Anal.154 (2000), 297–324. doi:10.1007/s002050000101.
4.
L.Cherfils and Y.Ilyason, On the stationary solutions of generalized reaction-diffusion equations with -Laplacian, Comm. Pure Appl. Anal.4 (2005), 9–22. doi:10.3934/cpaa.2005.4.9.
5.
M.Colombo and G.Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal.215 (2015), 443–496. doi:10.1007/s00205-014-0785-2.
6.
M.Colombo and G.Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal.218 (2015), 219–273. doi:10.1007/s00205-015-0859-9.
7.
M.Colombo and G.Mingione, Calderón-Zygmund estimates and non-uniformly elliptic operators, J. Funct. Anal.270 (2016), 1416–1478. doi:10.1016/j.jfa.2015.06.022.
8.
M.Filippakis and N.S.Papageorgiou, Multiple constant sign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Differential Equations245 (2008), 1883–1922. doi:10.1016/j.jde.2008.07.004.
9.
L.Gasinski and N.S.Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set Valued Anal.20 (2012), 417–423. doi:10.1007/s11228-011-0198-4.
10.
L.Gasinski and N.S.Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Problem Books in Math., Springer, Cham, 2016.
11.
L.Gasinski and N.S.Papageorgiou, Positive solutions for the Robin p-Laplacian problem with competing nonlinearities, Adv. Calc. Var.12 (2019), 31–56. doi:10.1515/acv-2016-0039.
12.
T.He, Z.Yao and Z.Sun, Multiple and nodal solutions for parametric Neumann problems with nonhomogeneous differential operator and critical growth, J. Math. Anal. Appl.449 (2017), 1133–1151. doi:10.1016/j.jmaa.2016.12.020.
13.
S.Hu and N.S.Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer Academic Publishers, Dordrecht, 1997.
14.
O.Ladyzhenskaya and N.Urallseva, Linear and Quasilinear Elliptic Equtions, Academic Press, New York, 1968.
15.
G.M.Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal.12 (1988), 1203–1219. doi:10.1016/0362-546X(88)90053-3.
16.
Z.Liu and Z.-Q.Wang, Schrödinger equations with concave and convex nonlinearities, Z. Angew. Math. Phys.56 (2005), 609–629. doi:10.1007/s00033-005-3115-6.
17.
P.Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J. Differential Equations90 (1991), 1–30. doi:10.1016/0022-0396(91)90158-6.
18.
N.S.Papageorgiou and V.D.Radulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim.69 (2014), 393–430. doi:10.1007/s00245-013-9227-z.
19.
N.S.Papageorgiou and V.D.Radulescu, Infinitely many nodal solutions for nonlinear nonhomogeneous Robin problems, Adv. Nonlin. Stud.16 (2016), 287–300.
20.
N.S.Papageorgiou, V.D.Radulescu and D.D.Repovs, Nodal solutions for nonlinear nonhomogeneous Robin problems, Rend. Lincei. Mat. Appl.29 (2018), 721–738.
21.
N.S.Papageorgiou, V.D.Radulescu and D.D.Repovs, Double-phase problems with reaction of arbitrary growth, Z. Angew. Math. Phys.69 (2018), 108. doi:10.1007/s00033-018-1001-2.
22.
N.S.Papageorgiou, V.D.Radulescu and D.D.Repovs, Nonlinear Analysis – Theory and Methods, Springer, Switzerland, 2019.
23.
N.S.Papageorgiou, V.D.Radulescu and D.D.Repovs, Double-phase problems and a discontinuity property of the spectrum, Proc. Amer. Math. Soc., In Press. doi:10.1090/proc/14466.
24.
P.Pucci and J.Serrin, The Maximum Principle, Birkhäser, Basel, 2007.
25.
V.D.Radulescu, Isotropic and anisotropic double-phase problems: Old and new, Opuscula Math.39 (2019), 259–279. doi:10.7494/OpMath.2019.39.2.259.
26.
Z.-Q.Wang, Nonlinear boundary value problems with concave nonlinearities near the origin, NoDEA Nonlinear Differential Equations Appl.8 (2001), 15–31. doi:10.1007/PL00001436.
27.
Q.Zhang and V.D.Radulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl.118 (2018), 159–203. doi:10.1016/j.matpur.2018.06.015.
28.
V.V.Zhikov, Averaging of functionals of the calculus of variations and elasticity, Math. USSR–Izv.29 (1987), 33–66. doi:10.1070/IM1987v029n01ABEH000958.
29.
V.V.Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci.173 (2011), 463–570. doi:10.1007/s10958-011-0260-7.
