## Existence of solutions with prescribed norm for semilinear elliptic equations.(English)Zbl 0877.35091

The nonlinear eigenvalue problem $-\Delta u(x) - g(u(x)) = \lambda u(x),\quad\lambda \in \mathbb{R},\;x \in \mathbb{R}^N$ is studied. It is supposed that $$g:\mathbb{R} \to \mathbb{R}$$ is continuous, odd, and such that $\alpha \int_0^s g(t) dt \leq g(s)s \leq \beta \int_0^s g(t) dt$ with some $$(2N+4)/N<\alpha\leq\beta< (2N)/(N-2)$$ if $$N\geq 3$$ or $$(2N+4)/N<\alpha\leq\beta$$ if $$n=1,2$$. A certain mild additional assumption is added in the case $$N=1$$. It is proved that for any $$c>0$$ there exists a weak solution $$u_c \in H^1(\mathbb{R}^N),\;\lambda_c \in \mathbb{R}$$ satisfying $$|u|_{L^2}= c,\;\lambda_c <0$$. The proof is based on a minimax approach used for the corresponding functional. Further, the dependence of $$|\nabla u_c|_{L^2}$$ and $$\lambda_c$$ on $$c$$ is described. Particularly, it follows $$|\nabla u_c|_{L^2} \to +\infty,\;\lambda_c \to -\infty$$ as $$c \to 0$$ and $$|\nabla u_c|_{L^2} \to 0,\;\lambda_c \to 0$$ as $$c \to +\infty$$.
Reviewer: M.Kučera (Praha)

### MSC:

 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations

### Keywords:

Palais-Smale sequence; Pohozaev identity; bifurcation
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