Nonlinear nonhomogeneous boundary value problems with competition phenomena

Du\v{s}an D. Repov\v{s}; Nikolaos S. Papageorgiou; Vicen\c{t}iu D. R\u{a}dulescu

arxiv: 1907.04999 · v1 · pith:BREBVI4Ynew · submitted 2019-07-11 · 🧮 math.AP

Nonlinear nonhomogeneous boundary value problems with competition phenomena

Nikolaos S. Papageorgiou , Vicen\c{t}iu D. R\u{a}dulescu , Du\v{s}an D. Repov\v{s} This is my paper

Pith reviewed 2026-05-24 23:24 UTC · model grok-4.3

classification 🧮 math.AP

keywords nonlinear boundary value problemsnonhomogeneous differential operatorscompeting nonlinearitiesmultiple solutionsMorse theorynodal solutionsvariational methodstruncation techniques

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The pith

For all small λ > 0, a nonlinear nonhomogeneous boundary value problem with competing superlinear and sublinear terms has at least five nontrivial smooth solutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates a boundary value problem driven by a nonhomogeneous differential operator that features competing nonlinearities: a superlinear convex term in the reaction and a sublinear concave term in the parametric boundary source. It establishes the existence of at least five nontrivial smooth solutions for all sufficiently small positive values of the parameter λ, consisting of four solutions of constant sign and one nodal solution. The work also constructs extremal constant sign solutions and analyzes their monotonicity and continuity with respect to λ. In the semilinear case, the analysis yields a sixth nontrivial solution without sign information. These results are obtained by combining variational methods, truncation and perturbation techniques, and Morse theory.

Core claim

We show that for all small parameter values λ>0, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter λ>0 varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.

What carries the argument

The nonhomogeneous differential operator with a superlinear reaction term competing against a sublinear parametric boundary term, whose associated energy functional is analyzed using variational methods and Morse theory to locate multiple critical points.

If this is right

Four constant sign solutions and one nodal solution exist for small λ.
Extremal constant sign solutions vary monotonically and continuously with λ.
A sixth solution appears in the semilinear case.
The truncation and perturbation techniques allow handling the competition between convex and concave terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach of balancing superlinear interior terms with sublinear boundary terms could apply to other classes of elliptic problems to produce multiplicity results.
Variational and Morse theoretic methods may reveal additional solutions or properties when the parameter range is extended beyond small values.
The existence of a nodal solution indicates that sign-changing behavior persists under the competition of nonlinearities.

Load-bearing premise

The boundary term is sublinear for small λ while the reaction is superlinear, creating the functional geometry needed for Morse theory to detect multiple critical points.

What would settle it

An explicit counterexample or numerical simulation for a specific small λ that yields only four or fewer nontrivial smooth solutions would falsify the result.

read the original abstract

We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $\lambda>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $\lambda>0$ varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper shows five solutions (four constant sign, one nodal) for small λ in nonhomogeneous BVP with competing terms via variational + Morse methods; the C^1 regularity issue for Morse theory is worth checking but does not sink the claim.

read the letter

The core result is that for small λ>0 the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal, plus extremal constant-sign solutions whose monotonicity and continuity in λ are tracked. In the semilinear case they add a sixth solution without sign information. The approach combines variational methods, truncation/perturbation, and Morse theory on the nonhomogeneous operator with superlinear reaction and sublinear boundary term. This is a direct extension of existing multiplicity techniques to the nonhomogeneous setting, and the extra work on extremal solutions and parameter dependence is useful detail. The methods are standard in the area and the abstract claim is internally consistent. The main soft spot is the stress-test point on regularity: nonhomogeneous operators typically produce C^1 but not C^2 energy functionals on W^{1,p}, and standard critical-group computations via Morse theory need either C^2 or a documented C^1 version with isolated critical points and a suitable pseudo-gradient flow. The paper description does not explicitly flag how this is handled, so a referee should ask for a short paragraph or reference confirming the version of Morse theory that applies. That is a technical clarification rather than a load-bearing flaw. The citation pattern is not visible from the abstract, but the result sits squarely inside the literature on competing nonlinearities. This is for people working on elliptic problems with nonstandard operators and multiplicity via variational methods. A reader already familiar with truncation arguments and Morse theory on C^1 functionals will get the most out of it. It is solid enough to send to referees; the central existence claim looks reachable with the stated tools once the regularity detail is confirmed.

Referee Report

2 major / 2 minor

Summary. The manuscript considers a nonlinear boundary value problem driven by a nonhomogeneous differential operator, with competing nonlinearities consisting of a superlinear convex reaction term and a sublinear concave parametric boundary term. It claims that for all sufficiently small λ > 0 the problem admits at least five nontrivial smooth solutions (four of constant sign and one nodal). In the semilinear case an additional sixth nontrivial solution is obtained. The approach relies on variational methods combined with truncation and perturbation techniques together with Morse theory; extremal constant-sign solutions are also constructed and their monotonicity/continuity properties with respect to λ are studied.

Significance. If the multiplicity result is rigorously established, the work would extend known multiplicity theorems from the semilinear setting to nonhomogeneous operators while incorporating competing convex-concave nonlinearities. The combination of truncation/perturbation with Morse theory for such operators, together with the analysis of extremal solutions, would constitute a substantive contribution to the literature on elliptic problems with variable growth.

major comments (2)

[Morse theory application / energy functional regularity] The central multiplicity claim (five solutions, including the nodal one) rests on Morse-theoretic computations of critical groups. For a nonhomogeneous operator of the form div(a(x,|∇u|)∇u) the associated energy functional is C¹ but in general not C² on W^{1,p}. Standard Morse inequalities and critical-group calculations require either C² regularity or a verified C¹ version with a well-defined pseudo-gradient flow and isolated critical points. The manuscript must explicitly address this regularity issue (e.g., by citing or deriving the appropriate C¹ Morse theory) because it is load-bearing for the nodal-solution count.
[Comparison between nonhomogeneous and semilinear cases] The abstract and approach description indicate that the semilinear case yields a sixth solution while the nonhomogeneous case stops at five; the distinction appears to hinge on the C² property available only in the semilinear setting. If the nonhomogeneous functional fails to be C², the truncation/perturbation argument must be shown to preserve the necessary isolation and index properties used in the Morse inequalities.

minor comments (2)

[Main theorem statement] Clarify the precise growth and convexity/concavity assumptions on the reaction and boundary terms in the statement of the main theorem to make the parameter range for λ explicit.
[Variational setting] Ensure that all references to the nonhomogeneous operator include the precise form of a(x,t) and verify that the functional satisfies the Palais-Smale condition under the stated hypotheses.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We provide point-by-point responses to the major comments below, and we are prepared to make revisions where necessary to address the concerns.

read point-by-point responses

Referee: [Morse theory application / energy functional regularity] The central multiplicity claim (five solutions, including the nodal one) rests on Morse-theoretic computations of critical groups. For a nonhomogeneous operator of the form div(a(x,|∇u|)∇u) the associated energy functional is C¹ but in general not C² on W^{1,p}. Standard Morse inequalities and critical-group calculations require either C² regularity or a verified C¹ version with a well-defined pseudo-gradient flow and isolated critical points. The manuscript must explicitly address this regularity issue (e.g., by citing or deriving the appropriate C¹ Morse theory) because it is load-bearing for the nodal-solution count.

Authors: We agree that the energy functional is C¹ but not necessarily C². Our approach relies on the C¹-Morse theory developed for functionals on Banach spaces, which requires the existence of a pseudo-gradient vector field and isolated critical points. The truncation and perturbation techniques employed in the paper ensure that the critical points under consideration are isolated, allowing the computation of critical groups. We will revise the manuscript to include an explicit reference to the relevant C¹ Morse theory literature and a short explanation of why the conditions are satisfied in our setting. This clarification will strengthen the presentation without altering the results. revision: yes
Referee: [Comparison between nonhomogeneous and semilinear cases] The abstract and approach description indicate that the semilinear case yields a sixth solution while the nonhomogeneous case stops at five; the distinction appears to hinge on the C² property available only in the semilinear setting. If the nonhomogeneous functional fails to be C², the truncation/perturbation argument must be shown to preserve the necessary isolation and index properties used in the Morse inequalities.

Authors: The additional sixth solution in the semilinear case is obtained through more detailed Morse index computations that are facilitated by the C² regularity. In the nonhomogeneous case, we obtain five solutions using the available C¹ theory. The truncation and perturbation are performed in such a way that the modified functionals coincide with the original one near the critical points of interest, thereby preserving the critical groups and isolation properties. We will add a remark in the revised version explaining this preservation more explicitly to address the concern. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; standard application of variational and Morse methods

full rationale

The paper proves existence of at least five solutions for small λ>0 via variational methods, truncation/perturbation, and Morse theory on the energy functional associated to the nonhomogeneous operator. No step reduces a claimed result to its own inputs by construction (no self-definitional relations, no fitted parameters presented as predictions). No load-bearing self-citation chains or uniqueness theorems imported from the authors' prior work are invoked to force the multiplicity count. The approach is self-contained against external benchmarks in critical point theory and Morse inequalities, which apply independently of this paper's specific results. The noted C^1 vs C^2 regularity issue for the functional is a potential technical gap in applicability but does not constitute circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from nonlinear analysis such as properties of the nonhomogeneous operator and the competing nonlinearities being superlinear and sublinear respectively.

axioms (2)

domain assumption The nonhomogeneous differential operator satisfies regularity and growth conditions suitable for variational formulation
Invoked to set up the energy functional and apply variational methods.
domain assumption The reaction term is superlinear and the boundary term is sublinear for the competition to produce multiple solutions
Central to the multiplicity result stated in the abstract.

pith-pipeline@v0.9.0 · 5668 in / 1208 out tokens · 26248 ms · 2026-05-24T23:24:42.346975+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

[1]

Aizicovici, N.S

S. Aizicovici, N.S. Papageorgiou, V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality constrai nts, Memoirs Amer. Math. Soc., Vol. 196, No. 905 (2008), pp. 70

work page 2008
[2]

Ambrosetti, H

A. Ambrosetti, H. Brezis, G. Cerami, Combined eﬀects of c oncave and convex nonlinearities in some elliptic problems, J. Functional Anal. 122 (1994), 519-543

work page 1994
[3]

Ambrosetti, P

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

work page 1973
[4]

Benci, P

V. Benci, P. D’Avenia, D. Fortunato, L. Pisani, Solitons in several space dimensions: Derrick’s problem and inﬁnitely many solutions, Arch. Rational Mech. Anal. 154 (2000), 297-324

work page 2000
[5]

Chang, Inﬁnite Dimensional Morse Theory and Multiple Solution Pro blems, Birkh¨ auser, Boston, 1993

K.C. Chang, Inﬁnite Dimensional Morse Theory and Multiple Solution Pro blems, Birkh¨ auser, Boston, 1993

work page 1993
[6]

Cherﬁls, Y

L. Cherﬁls, Y. Ilyasov, On the stationary solutions of ge neralized reaction diﬀusion equations with p&q Laplacian, Comm. Pure Appl. Anal. 4 (2005), 9-22

work page 2005
[7]

Cingolani, M

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

work page 2005
[8]

Corvellec, A

J.N. Corvellec, A. Hantoute, Homotopical stability of i solated critical points of continuous functionals, Set Valued Anal. 10 (2002), 143-164

work page 2002
[9]

Diaz, J.E

J.I. Diaz, J.E. Saa, Existence et unicit´ e de solutions p ositives pour certaines ´ equations ellip- tiques quasilin´ eaires,C. R. Acad. Sci. Paris S´ er. I 305 (1987), 521-524

work page 1987
[10]

Filippakis, N.S

M. Filippakis, N.S. Papageorgiou, Multiple constant s ign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Diﬀerential Equations 245 (2008), 1883-1922

work page 2008
[11]

Furtado, R

M. Furtado, R. Ruviaro, Multiple solutions for a semili near problem with combined terms and nonlinear boundary conditions, Nonlinear Anal. 74 (2011), 4820-4830

work page 2011
[12]

Garcia Azorero, I

J. Garcia Azorero, I. Peral Alonso, J. Rossi, A convex-c oncave problem with a nonlinear boundary condition, J. Diﬀerential Equations 198 (2004), 91-128

work page 2004
[13]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, Nonlinear Analysis , Chapman & Hall/CRC, Boca Raton, FL, 2006

work page 2006
[14]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, Existence and multipl icity of solutions for Neumann p- Laplacian type equations, Adv. Nonlinear Studies 8 (2008), 243-270

work page 2008
[15]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, A pair of positive solu tions for ( p, q)-equations with combined nonlinearities, Comm. Pure Appl. Anal. 13 (2014), 203-215

work page 2014
[16]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis , Springer, New York, 2016

work page 2016
[17]

Guedda, L

M. Guedda, L. V´ eron, Quasilinear elliptic equations i nvolving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879-902

work page 1989
[18]

S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory , Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997

work page 1997
[19]

S. Hu, N.S. Papageorgiou, Nonlinear Neumann problems w ith indeﬁnite potential and concave terms, Comm. Pure Appl. Anal. 14 (2015), 2561-2616

work page 2015
[20]

S. Hu, N.S. Papageorgiou, Elliptic equations with inde ﬁnite and unbounded potential and a nonlinear concave boundary condition, Commun. Contemp. Math. 19 (2017), no. 1, 1550090, 34 pp

work page 2017
[21]

Lieberman, The natural generalization of the natura l conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm

G. Lieberman, The natural generalization of the natura l conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Diﬀerential Equations 16 (1991), 311-361

work page 1991
[22]

Marano, S

S. Marano, S. Mosconi, N.S. Papageorgiou, Multiple sol utions to ( p, q)-Laplacian problems with resonant concave nonlinearity, Adv. Nonlin. Studies 16 (2016), 51-65

work page 2016
[23]

Motreanu, V

D. Motreanu, V. Motreanu, N.S. Papageorgiou, Topological and Variational Methods with Applications to Boundary Value Problems , Springer, New York, 2014

work page 2014
[24]

Mugnai, N.S

D. Mugnai, N.S. Papageorgiou, W ang’s multiplicity res ult for superlinear ( p, q)-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), 4919- 4937

work page 2014
[25]

Palais, Homotopy theory of inﬁnite dimensional mani folds, Topology 5 (1966), 1-16

R. Palais, Homotopy theory of inﬁnite dimensional mani folds, Topology 5 (1966), 1-16

work page 1966
[26]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Qualitative phen omena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393-430. NONLINEAR NONHOMOGENEOUS BOUNDARY V ALUE PROBLEMS 39

work page 2014
[27]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Resonant ( p, 2)-equations with asymmetric reaction, Anal. Appl. (Singap.) 13 (2015), 481-506

work page 2015
[28]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Multiple solutio ns with precise sign for nonlinear para- metric Robin problems, J. Diﬀerential Equations 256 (2014), 2449-2479

work page 2014
[29]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Coercive and nonc oercive nonlinear Neumann problems with indeﬁnite potential, Forum Math. 28 (2016), 545-571

work page 2016
[30]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Nonlinear nonhom ogeneous Robin problems with super- linear reaction term, Advanced Nonlinear Studies 16 (2016), 737-764

work page 2016
[31]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s, On a class of parametric ( p, 2)-equations, Appl. Math. Optim. 75 (2017), 193-228

work page 2017
[32]

Papageorgiou, P

N.S. Papageorgiou, P. Winkert, Resonant ( p, 2)-equations with concave terms, Applic. Anal. 94 (2015), 342-360

work page 2015
[33]

Pucci, J

P. Pucci, J. Serrin, The Maximum Principle , Birkh¨ auser, Basel, 2007

work page 2007
[34]

Sun, Multiplicity of solutions for a class of quasili near elliptic equations at resonance, J

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

work page 2012
[35]

M. Sun, M. Zhang, J. Su, Critical groups at zero and multi ple solutions for a quasilinear elliptic equation, J. Math. Anal. Appl. 428 (2015), 696-712. (Nikolaos S. Papageorgiou) National Technical University, Department of Mathemat- ics, Zografou Campus, Athens 15780, Greece & Institute of Ma thematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana...

work page 2015

[1] [1]

Aizicovici, N.S

S. Aizicovici, N.S. Papageorgiou, V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality constrai nts, Memoirs Amer. Math. Soc., Vol. 196, No. 905 (2008), pp. 70

work page 2008

[2] [2]

Ambrosetti, H

A. Ambrosetti, H. Brezis, G. Cerami, Combined eﬀects of c oncave and convex nonlinearities in some elliptic problems, J. Functional Anal. 122 (1994), 519-543

work page 1994

[3] [3]

Ambrosetti, P

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

work page 1973

[4] [4]

Benci, P

V. Benci, P. D’Avenia, D. Fortunato, L. Pisani, Solitons in several space dimensions: Derrick’s problem and inﬁnitely many solutions, Arch. Rational Mech. Anal. 154 (2000), 297-324

work page 2000

[5] [5]

Chang, Inﬁnite Dimensional Morse Theory and Multiple Solution Pro blems, Birkh¨ auser, Boston, 1993

K.C. Chang, Inﬁnite Dimensional Morse Theory and Multiple Solution Pro blems, Birkh¨ auser, Boston, 1993

work page 1993

[6] [6]

Cherﬁls, Y

L. Cherﬁls, Y. Ilyasov, On the stationary solutions of ge neralized reaction diﬀusion equations with p&q Laplacian, Comm. Pure Appl. Anal. 4 (2005), 9-22

work page 2005

[7] [7]

Cingolani, M

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

work page 2005

[8] [8]

Corvellec, A

J.N. Corvellec, A. Hantoute, Homotopical stability of i solated critical points of continuous functionals, Set Valued Anal. 10 (2002), 143-164

work page 2002

[9] [9]

Diaz, J.E

J.I. Diaz, J.E. Saa, Existence et unicit´ e de solutions p ositives pour certaines ´ equations ellip- tiques quasilin´ eaires,C. R. Acad. Sci. Paris S´ er. I 305 (1987), 521-524

work page 1987

[10] [10]

Filippakis, N.S

M. Filippakis, N.S. Papageorgiou, Multiple constant s ign and nodal solutions for nonlinear elliptic equations with the p-Laplacian, J. Diﬀerential Equations 245 (2008), 1883-1922

work page 2008

[11] [11]

Furtado, R

M. Furtado, R. Ruviaro, Multiple solutions for a semili near problem with combined terms and nonlinear boundary conditions, Nonlinear Anal. 74 (2011), 4820-4830

work page 2011

[12] [12]

Garcia Azorero, I

J. Garcia Azorero, I. Peral Alonso, J. Rossi, A convex-c oncave problem with a nonlinear boundary condition, J. Diﬀerential Equations 198 (2004), 91-128

work page 2004

[13] [13]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, Nonlinear Analysis , Chapman & Hall/CRC, Boca Raton, FL, 2006

work page 2006

[14] [14]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, Existence and multipl icity of solutions for Neumann p- Laplacian type equations, Adv. Nonlinear Studies 8 (2008), 243-270

work page 2008

[15] [15]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, A pair of positive solu tions for ( p, q)-equations with combined nonlinearities, Comm. Pure Appl. Anal. 13 (2014), 203-215

work page 2014

[16] [16]

Gasinski, N.S

L. Gasinski, N.S. Papageorgiou, Exercises in Analysis. Part 2: Nonlinear Analysis , Springer, New York, 2016

work page 2016

[17] [17]

Guedda, L

M. Guedda, L. V´ eron, Quasilinear elliptic equations i nvolving critical Sobolev exponents, Nonlinear Anal. 13 (1989), 879-902

work page 1989

[18] [18]

S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory , Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997

work page 1997

[19] [19]

S. Hu, N.S. Papageorgiou, Nonlinear Neumann problems w ith indeﬁnite potential and concave terms, Comm. Pure Appl. Anal. 14 (2015), 2561-2616

work page 2015

[20] [20]

S. Hu, N.S. Papageorgiou, Elliptic equations with inde ﬁnite and unbounded potential and a nonlinear concave boundary condition, Commun. Contemp. Math. 19 (2017), no. 1, 1550090, 34 pp

work page 2017

[21] [21]

Lieberman, The natural generalization of the natura l conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm

G. Lieberman, The natural generalization of the natura l conditions of Ladyzhenskaya and Uraltseva for elliptic equations, Comm. Partial Diﬀerential Equations 16 (1991), 311-361

work page 1991

[22] [22]

Marano, S

S. Marano, S. Mosconi, N.S. Papageorgiou, Multiple sol utions to ( p, q)-Laplacian problems with resonant concave nonlinearity, Adv. Nonlin. Studies 16 (2016), 51-65

work page 2016

[23] [23]

Motreanu, V

D. Motreanu, V. Motreanu, N.S. Papageorgiou, Topological and Variational Methods with Applications to Boundary Value Problems , Springer, New York, 2014

work page 2014

[24] [24]

Mugnai, N.S

D. Mugnai, N.S. Papageorgiou, W ang’s multiplicity res ult for superlinear ( p, q)-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc. 366 (2014), 4919- 4937

work page 2014

[25] [25]

Palais, Homotopy theory of inﬁnite dimensional mani folds, Topology 5 (1966), 1-16

R. Palais, Homotopy theory of inﬁnite dimensional mani folds, Topology 5 (1966), 1-16

work page 1966

[26] [26]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Qualitative phen omena for some classes of quasilinear elliptic equations with multiple resonance, Appl. Math. Optim. 69 (2014), 393-430. NONLINEAR NONHOMOGENEOUS BOUNDARY V ALUE PROBLEMS 39

work page 2014

[27] [27]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Resonant ( p, 2)-equations with asymmetric reaction, Anal. Appl. (Singap.) 13 (2015), 481-506

work page 2015

[28] [28]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Multiple solutio ns with precise sign for nonlinear para- metric Robin problems, J. Diﬀerential Equations 256 (2014), 2449-2479

work page 2014

[29] [29]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Coercive and nonc oercive nonlinear Neumann problems with indeﬁnite potential, Forum Math. 28 (2016), 545-571

work page 2016

[30] [30]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, Nonlinear nonhom ogeneous Robin problems with super- linear reaction term, Advanced Nonlinear Studies 16 (2016), 737-764

work page 2016

[31] [31]

Papageorgiou, V.D

N.S. Papageorgiou, V.D. R˘ adulescu, D.D. Repovˇ s, On a class of parametric ( p, 2)-equations, Appl. Math. Optim. 75 (2017), 193-228

work page 2017

[32] [32]

Papageorgiou, P

N.S. Papageorgiou, P. Winkert, Resonant ( p, 2)-equations with concave terms, Applic. Anal. 94 (2015), 342-360

work page 2015

[33] [33]

Pucci, J

P. Pucci, J. Serrin, The Maximum Principle , Birkh¨ auser, Basel, 2007

work page 2007

[34] [34]

Sun, Multiplicity of solutions for a class of quasili near elliptic equations at resonance, J

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

work page 2012

[35] [35]

M. Sun, M. Zhang, J. Su, Critical groups at zero and multi ple solutions for a quasilinear elliptic equation, J. Math. Anal. Appl. 428 (2015), 696-712. (Nikolaos S. Papageorgiou) National Technical University, Department of Mathemat- ics, Zografou Campus, Athens 15780, Greece & Institute of Ma thematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana...

work page 2015