Existence results for an anisotropic nonlocal problem involving critical and discontinuous nonlinearities

ABSTRACT

In this paper, we are interested in the existence of solutions to the anisotropic nonlocal problem $\begin{array}{}\text{{}(P{)}_{\delta }\text{}}& \begin{array}{rl}& -\sum _{i=1}^N\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right)={\left({\int }_{\mathrm{\Omega }}F(x,u)\right)}^{r}f(x,u)+\delta |u{|}^{p^{\ast }-2}u\text{in}\mathrm{\Omega },\\ & u\ge 0\text{in}\mathrm{\Omega },u=0\text{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where Ω is a smooth bounded domain of ${\mathbb{R}}^N$, $N\ge 2$, $\delta =0$ or $\delta =1,p_i$ and $r\ge 0$, $1<p_1\le \cdots \le p_N<p^{\ast }$ are parameters where $p^{\ast }=N\overline{p}/(N-\overline{p})$ is a critical exponent, $\overline{p}=N/\sum _{i=1}^{N}1/p_i$ and $\overline{p}<N$. The nonlinearity $f:\mathrm{\Omega }\times \mathbb{R}\to \mathbb{R}$ can be discontinuous, has subcritical growth subcritical growth and $F(x,t)={\int }_0^{t}f(x,s)\mathrm{d}s$. Under appropriate assumptions on f, we obtain the existence of nontrivial solutions for $(P{)}_{\delta }$ using the Ekeland's Variational Principle, Nonsmooth Mountain Pass Theorem and a Concentration Compactness-Principle.

COMMUNICATED BY:

• D. Repovš

KEYWORDS:

• Anisotropic operator
• local and nonlocal problem
• variational method
• Mountain Pass Theorem
• concentration compactness-principle

AMS SUBJECT CLASSIFICATIONS:

• 35A15
• 35J62

Disclosure statement

No potential conflict of interest was reported by the author(s).

ORCID

Leandro S. Tavares  http://orcid.org/0000-0003-0953-8037