Multiple solutions of degenerate perturbed elliptic problems involving a subcritical Sobolev exponent

Abstract

We study the degenerate
elliptic equation
$$
-\text{\rm div}(a(x)\nabla u)+b(x)u=
K(x)\vert u\vert ^{p-2}u+g(x)\quad \text{\rm in } \mathbb R^{N},
$$
where $N\geq 2$ and $2< p< 2^{*}$. We assume that $a\not\equiv 0$
is a continuous, bounded and nonnegative function,
while $b$ and $K$ are positive and essentially bounded
in $\mathbb R^{N}$. Under some assumptions on $a$, $b$ and $K$, which
control the location of zeros of $a$ and the behaviour of $a$, $b$ and
$K$ at infinity we prove that if the perturbation $g$ is sufficiently small
then the above problem has at least two distinct solutions in an
appropriate weighted Sobolev space. The proof relies essentially
on the Ekeland Variational Principle [< i> Nonconvex minimization problems< /i> , Bull. Amer. Math. Soc. < b> 1< /b> (1979),
443–473] and on the Mountain Pass
Theorem without the Palais-Smale condition, established in
Brezis-Nirenberg [< i> Positive solutions of nonlinear elliptic equations involving
critical Sobolev exponent< /i> , Comm. Pure Appl. Math. < b> 36< /b> (1983),
437–477], combined with a weighted variant of the
Brezis-Lieb Lemma [< i> A relation between pointwise convergence of functions and
convergence of functionals< /i> , Proc. Amer. Math. Soc. < b> 88< /b> (1983), 486–490],
in order to overcome the lack of compactness.