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

Abstract

We study the degenerate elliptic equation $$ -{\rm div}(a(x)\nabla u)+b(x)u= K(x)\vert u\vert ^{p-2}u+g(x)\quad \text{in } \mathbb R^{N}, $$ 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 [Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), 443–473] and on the Mountain Pass Theorem without the Palais-Smale condition, established in Brezis-Nirenberg [Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Comm. Pure Appl. Math. 36 (1983), 437–477], combined with a weighted variant of the Brezis-Lieb Lemma [A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490], in order to overcome the lack of compactness.