Existence of nontrivial solution for a quasilinear elliptic equation with (p, q)-Laplacian in $${\mathbb {R}}^N$$ involving critical Sobolev exponents

Abstract

We establish the existence of nontrivial weak solution for the nonlinear elliptic equation $$\begin{aligned} \left\{ \begin{array}{l} -\Delta _pu - \Delta _qu = \lambda f(x){\left| u \right| ^{k-2}}u + g(x){\left| u \right| ^{{p^*}-2}}u\quad \text {in} \quad {\mathbb {R}}^N\\ u(x) \ge 0 ; \quad x \in {\mathbb {R}}^N \end{array} \right. \end{aligned}$$ where $$2 \le q \le p< k< p^*: = \frac{Np}{N - p},p< N,0<\lambda $$ is a parameter. Moreover $$0 \le f(x) \in C\left( {\mathbb {R}}^N \right) \cap {L^r}\left( {\mathbb {R}}^N \right) $$ with $$r = \frac{p^*}{p^* - k}$$ and $$0 \le g(x) \in C\left( {\mathbb {R}}^N \right) \cap {L^\infty }\left( {\mathbb {R}}^N \right) $$ is considered. Also $$0<f(x)$$ is bounded on some open subset $$\Omega \subset {\mathbb {R}}^N$$ .