Pairs of nontrivial solutions for resonant Neumann problems

Trans. Amer. Math. Soc.

Rădulescu

2014

We examine semilinear Neumann problems driven by the Laplacian plus an unbounded and indefinite potential. The reaction is a Carathéodory function which exhibits linear growth near ±∞. We allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue, and we prove several multiplicity results. Our approach uses critical point theory, Morse theory and the reduction method (the Lyapunov-Schmidt method).

“…Then y n = 1 for all n 1, and so we may assume that (13) y n w −→ y in H 1 (Ω) and y n →y in L 2s (Ω) 1 s We set y n = u + n u + n , n 1.…”

Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential

Trans. Amer. Math. Soc.

Rădulescu

2014

“…Let û1 (β) ∈ C 1 (Ω) be the L 2 -normalized (that is, û1 (β) 2 = 1) principal eigenfunction. From (6) we see that it does not change sign and we choose it to be positive, that is, û1 (β) ∈ C + . Moreover, from Harnack's inequality (see Pucci and Serrin [14,p.…”

Section: Mathematical Backgroundmentioning

confidence: 99%

“…The following proposition is an easy consequence of the spectral properties of −Δu + β(z)u described above. See also Papageorgiou and Rȃdulescu [11] (Proposition 2.3) for (a) and Gasinski and Papageorgiou [6] for (b), (c).…”

Section: Mathematical Backgroundmentioning

confidence: 99%

Neumann problems with indefinite and unbounded potential and concave terms

Rădulescu

2015

Proc. Amer. Math. Soc.

We consider a semilinear parametric Neumann problem driven by the negative Laplacian plus an indefinite and unbounded potential. The reaction is asymptotically linear and exhibits a negative concave term near the origin. Using variational methods together with truncation and perturbation techniques and critical groups, we show that for all small values of the parameter the problem has at least five nontrivial solutions, four of which have constant sign.

“…It was pointed out by the referee that in mathematical biology, the Neumann model is a more realistic one. For some other recent results on nonlinear Neumann boundary value problems involving p-Laplacian, we refer to Gasiński and Papageorgiou [11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction

Gasiński

Bull. Malays. Math. Sci. Soc.

2015

Self Cite

We consider a generalized logistic equation driven by the Neumann pLaplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value λ * > 0 of the parameter, such that if λ > λ * , the problem has at least two positive solutions, if λ = λ * , the problem has at least one positive solution and it has no positive solution if λ ∈ (0, λ * ). Finally, we show that for all λ λ * , the problem has a smallest positive solution.