Unbalanced fractional elliptic problems with exponential nonlinearity: subcritical and critical cases

Abstract

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} \begin{array}{rllll} (-Δ)^{s_1}_{p}u+(-Δ)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \; \text{ in } \mb R^N, \end{array} \end{equation*} \noi where $1<q<p$, $0<s_2\le s_1<1$, $p{s}_1=N$, $\ba\in[0,N)$, and $V,K:\mb R^N\to\mb R$, $f:\mb R\to \mb R$ are continuous functions satisfying some natural hypotheses. We are concerned both with the case when $f$ has a subcritical growth and with the critical framework with respect to the exponential nonlinearity. By combining a Moser-Trudinger type inequality for fractional Sobolev spaces with Schwarz symmetrization techniques and related variational methods, we prove the existence of nonnegative solutions.

arXiv commentThere are some improved and complete results in this version

References in corpus (6)

• Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves
• Double-phase problems and a discontinuity property of the spectrum
• Fractional Adams-Moser-Trudinger type inequalities
• Existence and multiplicity of solutions for double-phase Robin problems
• Existence and multiplicity of solutions for fractional Schrödinger-Kirchhoff equations with Trudinger-Moser nonlinearity
• On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces