On a nonhomogeneous and singular quasilinear equation involving critical growth inR2

Soliton solutions for a class of Schrödinger equations with a positive quasilinear term and critical growth

“…In [15], the authors have considered quasilinear elliptic equations of the second order with critical exponential growth of the form Using the dual approach, it was proved the existence of positive solutions for (1.4), under the conditions ()–() and for nonlinearities with critical exponential growth, which contain the following one as an example: with for and , (see similar results in [10, 14, 15, 27]). From these works, the maximal growth to treat variationally problems like (1.4) is of the form when the parameter is negative.…”

“…To study the case when the parameter is negative, in [8], the authors adopted the dual approach by working in the usual Sobolev space environment. For that, they performed the change of variables: , where is defined by Applying this change of variables, the authors reduced () to the semilinear equation Along this line, still in the case when the parameter is negative, we refer to [2, 10, 13–15, 27, 33], for works on the existence and multiplicity of solutions for () in the critical growth range.…”

Solutions for a class of singular quasilinear equations involving critical growth in R2$\mathbb {R}^2$

“…Results of existence, multiplicity, asymptotic behavior and concentration of solutions for quasilinear problems of type (1.1), involving critical growth in the Sobolev case, have been studied in a large number of works, we refer the reader for instance to [17, 25, 30, 31, 39] and references therein. Quasilinear equations involving subcritical or critical exponential growth in $$\mathbb {R}^2$$ were studied by various authors, see [2, 13, 16, 18, 19, 21, 34, 42]. For the singular case in $$\mathbb {R}^2$$ with $$\beta =1$$ and the equation involving a non‐homogeneous term, the authors in [13] prove the existence of two solutions by applying minimax methods.…”

“…Quasilinear equations involving subcritical or critical exponential growth in $$\mathbb {R}^2$$ were studied by various authors, see [2, 13, 16, 18, 19, 21, 34, 42]. For the singular case in $$\mathbb {R}^2$$ with $$\beta =1$$ and the equation involving a non‐homogeneous term, the authors in [13] prove the existence of two solutions by applying minimax methods. For more general quasilinear problems involving critical growth in the Sobolev case, see for example [12, 22].…”

“…According to some compactness results involving the space X and by exploring the properties of the change of variables, estimates related to the mountain‐pass level of I and a singular Trudinger–Moser inequality, we show that the functional I has a nontrivial critical point at the mountain‐pass level, which is a positive solution of (1.1). Here, we do not use the structure of Orlicz spaces as explored in the article [13], because we would have an upper limit on β, namely, $$\beta \le 1$$. This restriction would be necessary to prove the convexity of the functional $$\Phi (v)=\int _{\mathbb {R}^2}V(x) f^2(v) \, \mathrm{d}x$$, which was essential in the arguments used to show the Palais–Smale condition in an appropriate interval.…”