Multiplicity theorems for resonant and superlinear nonhomogeneous elliptic equations

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu



We consider nonlinear elliptic equations driven by the sum of a $p$-Laplacian ($p> 2$) and a Laplacian. We consider two distinct cases. In the first one, the reaction $f(z,\cdot)$ is $(p-1)$-linear near $\pm\infty$ and resonant with respect to a nonprincipal variational eigenvalue of $(-\Delta_{p},W_{0}^{1,p}(\Omega))$. We prove a multiplicity theorem producing three nontrivial solutions. In the second case, the reaction $f(z,\cdot)$ is $(p-1)$-superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. We prove two multiplicity theorems. In the first main result we produce six nontrivial solutions all with sign information and in the second theorem we have five nontrivial solutions. Our approach uses variational methods combined with the Morse theory, truncation methods, and comparison techniques.


Resonance; multiple solution; superlinear reaction; nodal solutions; critical groups

Full Text:



S. Aizicovici, N.S. Papageorgiou and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Memoirs Amer. Math. Soc. Vol 196, No. 915 (2008).

S. Aizicovici, N.S. Papageorgiou and V. Staicu, On p-superlinear equations with a nonhomogeneous differential operator, Nonlinear Differential Equations Appl. (NoDEA) 20 (2013), 151–175.

A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Anal. 14 (1973), 349–381.

D. Arcoya and D. Ruiz, The Ambrosetti–Prodi problem for the p-Laplace operator, Comm. Partial Differential Equations 31 (2006), 849–865.

T. Bartsch, Z. Liu and T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc. 91 (2005), 129–152.

R. Benguria, H. Brezis and E.H. Lieb, The Thomas–Fermi–von Weizsäcker theory of atoms and molecules, Comm. Math. Physics 79 (1981), 167–180.

S. Cingolani and M. Degiovanni, Nontrivial solutions for p-Laplace equations with right hand side having p-linear growth, Comm. Partial Differential Equations 30 (2005), 1191–1203.

S. Cingolani and G. Vannella, Critical groups computations on a class of Sobolev Banach spaces via Morse index, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), 271–292.

M. Degiovanni and S. Lancelotti, Linking over cones and nontrivial solutions for pLaplace equations with p-superlinear nonlinearity, Ann. Inst. H. Poincaré Analyse NonLinéaire 24 (2007), 907-919.

J.I. Diaz and J.E. Saa, Existence et unicité de solutions positives pour certaines équations elliptiques quasilinéaires, C.R. Acad. Sci. Paris Sér. 305 (1987), 521–524.

J. Dugundji, Topology, Allyn and Bacon Inc, Boston (1966).

N. Dunford and J. Schwartz, Linear Operators I. General Theory, Wiley–Interscience, New York (1958).

E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions in Hamiltonian systems, Invent. Math. 45 (1978), 139–174.

M. Filippakis, A. Kristaly and N.S. Papageorgiou, Existence of five nonzero solutions with constant sign for a p-Laplacian equation, Discrete Cont. Dyn. Systems 24 (2009), 405–440.

L. Gasinski and N.S. Papageorgiou, Nonlinear Analysis, Chapman, Hall/CRC, Boca Raton, Fl. (2006).

L. Gasinski and N.S. Papageorgiou, Multiple solutions for nonlinear coercive problems with a nonhomogeneous differential operator and a nonsmooth potential, Set. Valued Var. Anal. 20 (2012), 417–443.

L. Iturriaga, E. Massa, J. Sachez and P. Ubilla, Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations 248 (2010), 309–327.

O. Ladyzhenskaya and N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York (1968).

G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal. 12 (1988), 1203–1219.

N.S. Papageorgiou and S. Kyritsi, Handbook of Applied Analysis, Springer, New York (2009).

N.S. Papageorgiou and V.D. Rădulescu, Qualitative phenomena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393–430.

N.S. Papageorgiou and G. Smyrlis, On nonlinear nonhomogeneous resonant Dirichlet equations, Pacific J. Math. 264 (2013), 421–453.

P. Pucci and J. Serrin, The Maximum Principle, Birkhäuser, Basel (2007).

M. Sun, Multiplicity of solutions for a class of quasilinear elliptic equations at resonance, J. Math. Anal. Appl. 386 (2012), 661–668.


  • There are currently no refbacks.

Partnerzy platformy czasopism