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

DOI: http://dx.doi.org/10.12775/TMNA.2000.021

#### 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.

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.

#### Keywords

Degenerate elliptic problem; weighted Sobolev space; unbounded domain; perturbation; multiple solutions

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