Elliptic problem involving logarithmic weight under exponential nonlinearities growth

Abstract

In this paper, we study the nonlinear weighted elliptic problem where B is the unit ball of , , and the singular logarithm weight with the limiting exponent in the Trudinger–Moser embedding. The nonlinearities are critical or subcritical growth in view of Trudinger–Moser inequalities. We prove the existence of nontrivial solutions via the critical point theory. In the critical case, the associated energy functional does not satisfy the compactness condition. We give a new growth condition and we point out its importance for checking the Palais–Smale compactness condition.

Least energy nodal solutions for a weighted $(N, p)$-Schrödinger problem involving a continuous potential under exponential growth nonlinearity

“…To be more specific, the authors demonstrated the existence of positive solutions for the above problem by combining variational techniques with regularity arguments. Alternatively, [4,10,11,14,19,20,43] investigated elliptic equations with weighted N-Laplacian operator and critical Trudinger-Moser nonlinearities, while this paper will focus on a different class of problems.…”