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Unique positive solution of semilinear elliptic equations involving concave and convex nonlinearities in \(\mathbb R^N\). (English) Zbl 1354.35047

Summary: In this article, we investigate the effect of the coefficient \( a(z) \) on the existence of positive solution of the subcritical semilinear elliptic problem. We prove for sufficiently large \( \lambda,\mu > 0\), there exists at least one positive solution for the problem \[ - \Delta v +\mu b(z) v = a(z) v^{p-1} +\lambda h(z)v^{q-1} \] where \(v \in H^{1}(\mathbb{R^{N}})\), \(1\leq q < 2 < p < 2^{*} =\dfrac{2N}{(N-2)}\) for \(N \geq 3\).

MSC:

35J61 Semilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B09 Positive solutions to PDEs
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References:

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