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DCDS

In this paper, arbitrarily many solutions, in particular arbitrarily
many nodal solutions, are proved to exist for perturbed elliptic
equations of the form
\begin{equation*}\label{}
\left\{
\begin{array}{ll}
\displaystyle -\Delta_p u+|u|^{p-2}u = Q(x)(f(u)+\varepsilon g(u)),\ \ \ x\in \mathbb R^N,
\\
u\in W^{1,p}(\mathbb R^N),
\end{array}
\right. (P_\varepsilon)
\end{equation*}
where $\Delta_p$ is the $p$-Laplacian operator defined by $\Delta_p
u=\text{div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $Q\in \mathcal{C}(\mathbb
R^N,\mathbb R)$ is a positive function, $f\in\mathcal{C}(\mathbb R, \mathbb
R)$ oscillates either near the origin or near the infinity, and
$\epsilon$ is a real number. For $g$ it is only required that
$g\in\mathcal{C}(\mathbb R, \mathbb R)$. Under appropriate assumptions on
$Q$ and $f$ the following results which are special cases of more
general ones are proved: the unperturbed problem $(P_0)$ has
infinitely many nodal solutions, and for any $n\in\mathbb N$ the
perturbed problem $(P_\varepsilon)$ has at least $n$ nodal solutions
provided that $|\epsilon|$ is sufficiently small.

DCDS

In this paper we study the
existence and multiplicity of nontrivial solutions for semilinear elliptic resonance problems with
a bounded nonlinearity.

DCDS

In this
paper, existence and multiplicity of nontrivial solutions are
obtained for some nonlinear elliptic boundary value problems with
perturbation terms of arbitrary growth. Results are obtained via
variational arguments.

keywords:
minimax methods
,
nontrivial solution.
,
multiplicity
,
perturbation term
,
critical group
,
Elliptic equation
,
resonance

DCDS

Existence of a solution of the nonlinear Schrödinger system
\begin{equation*}
\left\{ \begin{aligned}
& - \Delta u + V_1(x) u=\mu_1(x) u^3 + \beta(x) u v^2 \qquad\mbox{in}\ \mathbb{R}^N, \\
& - \Delta v + V_2(x) v=\beta(x) u^2 v + \mu_2(x) v^3 \qquad \mbox{in}\ \mathbb{R}^N, \\
& u>0,\ v>0,\quad u,\ v\in H^1(\mathbb{R}^N),
\end{aligned} \right.
\end{equation*}
where $N=1,2,3$, and $V_j,\mu_j,\beta$ are continuous functions of
$x\in\mathbb{R}^N$, is proved provided that either $V_j,\mu_j,\beta$ are
invariant under the action of a finite subgroup of $O(N)$ or there
is no such invariance assumption. In either case the result is
obtained both for $\beta$ small and for $\beta$ large in terms of
$V_j$ and $\mu_j$.

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