Calc. Var.
DOI 10.1007/s00526-014-0788-8 Calculus of Variations
Superlinear critical resonant problems with small forcing
term
Mabel Cuesta · Colette De Coster
Received: 15 May 2014 / Accepted: 10 October 2014
© Springer-Verlag Berlin Heidelberg 2014
Abstract We prove the existence of solutions of a class of quasilinear elliptic problems with
Dirichlet boundary conditions of the following form
Lu = g(u ) - f in ,
u ∈ X ,
where ⊂ R
N
is a bounded domain, N ≥ 2, the differential operator is Lu =
-div(|∇u |
p-2
∇u ) - λ
1
|u |
p-2
u with X = W
1, p
0
() or Lu =
2
u - λ
1
u with X = H
2
0
(),
the nonlinearity is given by g(u ) = (u
+
)
q
or g(u ) =|u |
q -1
u i.e. it is a superlinear, at most
critical, term and f is a small reaction term. We give an abstract formulation for which solu-
tions are found by minimization on an appropriate subset of the Nehari manifold associated
to our problem. Our method can be also applied to other related problems involving indefinite
weights.
Mathematics Subject Classification 35J20 · 35J70 · 35P05 · 35P30
Communicated by A. Malchiodi.
M. Cuesta (B )
Université du Littoral Côte d’Opale, LMPA, FR CNRS 2956,
50, rue F. Buisson, B.P. 699, 62228 Calais, France
e-mail: Mabel.Cuesta@lmpa.univ-littoral.fr
C. De Coster
Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956,
Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9, France
e-mail: Colette.DeCoster@univ-valenciennes.fr
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