Acta Applicandae Mathematicae ( IF 1.2 ) Pub Date : 2021-05-07 , DOI: 10.1007/s10440-021-00412-7 Uberlandio B. Severo , Diogo de S. Germano
In this work, we study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations:
$$ -\mbox{div} (g^{2}(u)\nabla u) + g(u)g'(u)|\nabla u|^{2} + V(x)u = f(x,u) + h(x)g(u) \quad \mbox{in}\quad \mathbb{R}^{N}, $$where \(N\geq 3\), \(g:\mathbb{R}\rightarrow \mathbb{R}_{+}\) is a continuously differentiable function, \(V(x)\) is a potential that can change sign, the function \(h(x)\) belongs to \(L^{2N/(N+2)}(\mathbb{R}^{N})\) and the nonlinearity \(f(x,s)\) is possibly discontinuous and may exhibit critical growth. In order to obtain the nonexistence result, we deduce a Pohozaev identity and the existence of solution is proved by means of a fixed point theorem.
中文翻译：
一类具有临界增长的拟线性Schrödinger方程解的存在性和不存在性。
在这项工作中，我们研究以下一类拟线性Schrödinger方程解的存在性和不存在性：
$$-\ mbox {div}（g ^ {2}（u）\ nabla u）+ g（u）g'（u）| \ nabla u | ^ {2} + V（x）u = f（x ，u）+ h（x）g（u）\ quad \ mbox {in} \ quad \ mathbb {R} ^ {N}，$$其中\（N \ geq 3 \），\（g：\ mathbb {R} \ rightarrow \ mathbb {R} _ {+} \）是一个连续可微的函数，\（V（x）\）是一个可以改变符号，函数\（h（x）\）属于\（L ^ {2N /（N + 2）}（\ mathbb {R} ^ {N}）\）和非线性\（f（x ，s）\）可能是不连续的，并且可能会出现临界增长。为了获得不存在的结果，我们推导了一个Pohozaev恒等式，并通过不动点定理证明了解的存在性。
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