In this article, we study the multiplicity of solutions for a class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$. Under the appropriate assumption, we prove that there are at least two solutions for the equation by Nehari manifold and Ekeland variational principle, one of which is the ground state solution.
In this paper, we study the following generalized quasilinear
Schrödinger equation
−
div
(
g
2
(
u
)
∇
u
)
+
g
(
u
)
g
′
(
u
)
|
∇
u
|
2
+
V
(
x
)
u
=
λ
f
(
x
,
u
)
+
h
(
x
,
u
)
,
x
∈
R
N
,
where
λ
>
0
,
N
≥
3
,
g
∈
C
1
(
R
,
R
+
)
. By using a change of
variable, we obtain the existence of positive solutions for this problem
with concave and convex nonlinearities via the Mountain Pass Theorem. Our
results generalize some existing results.
Abstract
We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian with $$1<q<p$$
1
<
q
<
p
. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.
In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole [Formula: see text] with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev embeddings.
In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: [Formula: see text] where [Formula: see text] are two parameters, [Formula: see text], [Formula: see text] is a potential well, [Formula: see text] and [Formula: see text]. Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on [Formula: see text] due to [Formula: see text]. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.
In this paper, we study the quasilinear Schrödinger equation involving concave and convex nonlinearities. When the pair of parameters belongs to a certain subset of ℝ2, we establish the existence of a nontrivial mountain pass-type solution and infinitely many negative energy solutions by using some new techniques and dual fountain theorem. Recent results from the literature are improved and extended.
A class of fourth-order elliptic equations with concave and convex nonlinearities in $\mathbb{R}^N$
