## Existence of positive solutions of the equation $$-\Delta u+a(x)u=u^{(N+2)/(N-2)}$$ in $${\mathbb{R}}^ N$$.(English)Zbl 0705.35042

Consider the following problem: find u such that $(*)\;u(x)>0,\;-\Delta u+a(x)u=| u|^{2^*-2}u\text{ on } \Omega,\;u\in {\mathcal D}^{1,2}(\Omega),$ where $$\Omega \subseteq {\mathbb{R}}^ N$$, $$N\geq 3$$, $$2^*=2N/(n-2)$$, $$a(x)\geq 0$$ and $${\mathcal D}^{1,2}(\Omega)$$ is a closure of $$C^{\infty}_ 0(\Omega)$$ with respect to the standard Sobolev norm of $$H^ 1(\Omega)$$. Then, they prove the following Theorem. If $$\Omega ={\mathbb{R}}^ N$$ and $$a(x)\geq 0\forall x\in {\mathbb{R}}^ N$$ and $$a(x)\geq \nu >0$$ in a neighborhood of a point $$\bar x,$$ $$\exists p_ 1<N/2$$ and $$p_ 2>N/2$$ and for $$N=3$$, $$p_ 2<3$$, such that $$a(x)\in L^ p\forall p\in [p_ 1,p_ 2]$$, $$| a|_{L^{N/2}}<S(2^{2/N}-1)$$, where $S=\inf \{\int_{{\mathbb{R}}^ N}| \nabla u|^ 2 dx/(\int_{{\mathbb{R}}^ N}| u|^{2^*} dx)^{2/2^*};\;u\in {\mathcal D}^{1,2}({\mathbb{R}}^ N)\},$ then the problem (*) has at least one positive solution.
Reviewer: Y.Ebihara

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations

### Keywords:

critical value; positive solution
Full Text:

### References:

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