Nontrivial solution of nonlinear scalar field equations with strong nonlinearity. (English) Zbl 0677.35036
Summary: We use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions of the following scalar field equations with strong nonlinearity
\[
- \sum^{N}_{i=1}\partial /\partial x_ i(| \nabla u|^{p- 2}(\partial u/\partial x_ i))+a(x)| u|^{q-2} u=f(x,u),\quad x\in R^ N,\quad N\geq 2,
\]
\[ u\in E\equiv \{u\in L^ q(R^ N)| \quad u\quad real,\quad \partial u/\partial x_ i\in L^ p(R^ N),\quad 1\leq i\leq N\}, \] where \(2\leq p\leq q\), \(p=N\).
\[ u\in E\equiv \{u\in L^ q(R^ N)| \quad u\quad real,\quad \partial u/\partial x_ i\in L^ p(R^ N),\quad 1\leq i\leq N\}, \] where \(2\leq p\leq q\), \(p=N\).
MSC:
35J60 | Nonlinear elliptic equations |
35D05 | Existence of generalized solutions of PDE (MSC2000) |
35J20 | Variational methods for second-order elliptic equations |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
35A35 | Theoretical approximation in context of PDEs |