\((p,N)\) equations with critical exponential nonlinearities in \(\mathbb{R}^N\)
In this paper the authors address the existence of nontrivial solutions for \((p,N)\) equations in \(\mathbb{R}^N\) with critical exponential growth. An application of a Brezis and Lieb type lemma for exponential nonlinearities also leads to a multiplicity result. More precisely, the authors consider the following equation in \(\mathbb{R}^N\): \[ -\Delta_p u-\Delta_N u+|u|^{p-2}u+|u|^{N-2}u=\lambda h(x)u_+^{q-1}+\gamma f(x,u), \] where \(N\ge 2\), \(\lambda, \gamma>0\), \(1<p<N<\infty\), \(1<q<N\), and \(h\in L^{\theta}(\mathbb{R}^N)\) is a positive function with \(\theta=\frac{N}{N-q}\). It is assumed that \(f\) is of exponential type, is a Carathéodory function, with \(f(\cdot, u)=0\) for \(u\le 0\), and \begin{itemize} \item \(\exists \alpha_0>0\) such that \(\forall \varepsilon>0\), \(\exists \kappa_{\varepsilon}>0\) with \[ f(x,u)\le\varepsilon u^{N-1}+\kappa_{\varepsilon}(e^{\alpha_0u^{N'}}-S_{N-2}(\alpha_0,u)) \] for a.e. \(x\in\mathbb{R}^N\) and all \(u\in\mathbb{R}_0^+\). Here \[ N'=\frac{N}{N-1}, \qquad S_{N_2}(\alpha_0,u)=\sum_{j=0}^{N-2}\frac{\alpha_0^ju^{jN'}}{j!}. \] \item \(\exists \nu>N\) with \(0<\nu \int_0^uf(x,v)dv \le uf(x,u)\) for a.e. \(x\in\mathbb{R}^N\) and any \(u\in\mathbb{R}^+\). \end{itemize} The first main result of the paper proves the existence of a nontrivial nonnegative solution of the equation considered. More precisely, the authors show that there exists \(\tilde{\lambda}>0\) such that the equation admits at least one nontrivial nonnegative solution \(u_{\lambda,\gamma}\in W^{1,p}(\mathbb{R}^N)\cap W^{1,N}(\mathbb{R}^N)\) for all \(\lambda\in(0,\tilde{\lambda})\), and for all \(\gamma>0\). Furthermore, \[ \lim\limits_{\lambda\rightarrow 0+}||u_{\lambda,\gamma}||=0. \] The key ingredient of the proof of this result is the Ekeland variational principle. The second main result of this paper establishes a multiplicity result, under two additional assumptions on \(f\). Under those conditions, the authors prove there exists \(\gamma^*>0\) such that for all \(\gamma>\gamma^*\), there exists \(\hat{\lambda}=\hat{\lambda}(\gamma)>0\) so that the equation considered has a nontrivial nonnegative solution \(v_{\lambda,\gamma}\) for all \(\lambda\in (0,\hat{\lambda}]\). Additionally, if \(\lambda<\min\{\tilde{\lambda},\hat{\lambda}\}\), then \(v_{\lambda,\gamma}\) is a second solution of the equation, independent of the solution \(u_{\lambda,\gamma}\) obtained in their first main result. The second solution is constructed by a mountain pass argument at a suitable small and positive energy level. Another main ingredient in the proof is a Brezis and Lieb type lemma for exponential nonlinearities.
scientific article; zbMATH DE number 7358789
scientific article