Concentration of solutions for (N,q)-Laplacian equation with Trudinger–Moser nonlinearity

Wang, Li and Wang, Jun and Zhang, Binlin (2023) Concentration of solutions for (N,q)-Laplacian equation with Trudinger–Moser nonlinearity. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (14). pp. 1-32. ISSN 1417-3875

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Abstract

In this article, we consider the concentration of positive solutions for the following equation with Trudinger–Moser nonlinearity: −∆Nu−∆qu+V(εx)(|u|N−2u+|u|q−2u) = f(u), x ∈ RN, u ∈W1,N(RN)∩W1,q(RN), x ∈RN, where V is a positive continuous function and has a local minimum, ε > 0 is a small parameter, 2 ≤ N < q < +∞, f is C1 with subcritical growth. When V and f satisfy some appropriate assumptions, we construct the solution uε that concentrates around any given isolated local minimum of V by applying the penalization method for the above equation.

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